#$&* course MTH 279 Part I: The equation m x '' = - k x*********************************************
.............................................
Given Solution: If x = cos(t) then x ' = - sin(t) and x '' = - cos(t). Substituting the expressions for x and x '' into the equation we obtain m * (-cos(t)) = - k * cos(t). Dividing both sides by cos(t) we obtain m = k. If m = k, then the equation is satisfied. If m is not equal to k, it is not. If x = sin(sqrt(k/m) * t) then x ' = sqrt(k / m) cos(sqrt(k/m) * t) and x '' = -k / m sin(sqrt(k/m) * t). Substituting this into the equation we have m * (-k/m sin(sqrt(k/m) * t) ) = -k sin(sqrt(k/m) * t Simplifying both sides we see that the equation is true. The same procedure can and should be used to show that the third equation is true, while the fourth is not. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q002. An incorrect integration of the equation x ' = 2 x + t yields x = x^2 + t^2 / 2. After all the integral of x is x^2 / 2 and the integral of t is t^2 / 2. Show that substituting x^2 + t^2 / 2 (or, if you prefer to include an integration constant, x^2 + t^2 / 2 + c) for x in the equation x ' = 2 x + t does not lead to equality. Explain what is wrong with the reasoning given above. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 2x+t ≠ 2 (x^2 + t^2/2) + t The equation was integrated with respect to two different variables (x and t, respectively), as opposed to a single variable, which is incorrect. When differentiated correctly (with respect to a single variable), the equation does not lead to equality when substituted and evaluated. Confidence rating: 3
.............................................
Given Solution: The given function is a solution to the equation, provided its derivative x ' satisfies x ' = 2 x + t. It would be tempting to say that the derivative of x^2 is 2 x, and the derivative of t^2 / 2 is t. The problem with this is that the derivative of x^2 was taken with respect to x and the derivative of t^2 / 2 with respect to t. We have to take both derivatives with respect to the same variable. Similarly we can't integrate the expression 2 x + t by integrating the first term with respect to x and the second with respect to t. Since in this context x ' represent the derivative of our solution function x with respect to t, the variable of integration therefore must be t. We will soon see a method for solving this equation, but at this point we simply cannot integrate our as-yet-unknown x(t) function with respect to t. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q003. The general solution to the equation m x '' = - k x is of the form x(t) = A cos(omega * t + theta_0), where A, omega and theta_0 are constants. (There are reasons for using the symbols omega and theta_0, but for right now just treat these symbols as you would any other constant like b or c). Find the general solution to the equation 5 x'' = - 2000 x: Substitute A cos(omega * t + theta_0) for x in the given equation. The value of one of the three constants A, omega and theta_0 is dictated by the numbers in the equation. Which is it and what is its value? One of the unspecified constants is theta_0. Suppose for example that theta_0 = 0. What is the remaining unspecified constant? Still assuming that theta_0 = 0, describe the graph of the solution function x(t). Repeat, this time assuming that theta_0 = 3 pi / 2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If x(t) = Acos(ωt+theta_0), then x(t) = -A ωsin(ωt+theta_0), and x(t) = -A ω^2cos(ωt+theta_0). We can plug both of these back into the original equation (mx = -kx). Doing so yields -mAω^2cos(ωt+theta_0) = -kAcos(ωt+theta_0). The Acos(ωt+theta_0) divides out, yielding -m ω^2 = -k. Solving for ω, we get ω = sqrt(k/m). The value of ω by the other given equation (5x = -2000x) is ω = sqrt(2000/5) = 20. In the third part, we assume that theta_0 = 0. The remaining unspecified constant at this point is A, which can be any real number, and the same can be said for theta_0. The graph of the function is a cosine wave, with a peak amplitude of A (full peak to peak amplitude of 2A). The period of this function is pi/10 rads. The general solution is x(t) = Acos(20t+theta_0) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: If x = A cos(omega * t + theta_0) then x ' = - omega A sin(omega * t + theta_0) and x '' = -omega^2 A cos(omega * t + theta_0). Our equation therefore becomes m * (-omega^2 A cos(omega * t + theta_0) ) = - k A cos(omega * t + theta_0). Rearranging we obtain -m omega^2 A cos(omega * t + theta_0) = -k A cos(omega * t + theta_0) so that -m omega^2 = - k and omega = sqrt(k/m). Thus the constant omega is determined by the equation. The constants A and theta_0 are not determined by the equation and can therefore take any values. No matter what values we choose for A and theta_0, the equation will be satisfied as long as omega = sqrt(k / m). Our second-order equation m x '' = - k x therefore has a general solution containing two arbitrary constants. In the present equation m = 5 and k = 2000, so that omega = sqrt(k / m) = sqrt(2000 / 5) = sqrt(400) = 20. Our solution x(t) = A cos(omega * t + theta_0) therefore becomes x(t) = A cos(20 t + theta_0). If theta_0 = 0 the function becomes x(t) = A cos( 20 t ). The graph of this function will be a 'cosine wave' with a 'peak' at the origin, and a period of pi / 10. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q004. In the preceding equation we found the general solution to the equation 5 x'' = - 2000 x. Assuming SI units, this solution applies to a simple harmonic oscillator of mass 5 kg, which when displaced to position x relative to equilibrium is subject to a net force F = - 2000 N / m * x. With these units, sqrt(k / m) has units of sqrt( (N / m) / kg), which reduce to radians / second. Our function x(t) describes the position of our oscillator relative to its equilibrium position. Evaluate the constants A and theta_0 for each of the following situations: The oscillator reaches a maximum displacement of .3 at clock time t = 0. The oscillator reaches a maximum displacement of .3 , and at clock time t = 0 its position is x = .15. The oscillator has a maximum velocity of 2, and is at its maximum displacement of .3 at clock time t = 0. The oscillator has a maximum velocity of 2, which occurs at clock time t = 0. (Hint: The velocity of the oscillator is given by the function x ' (t) ). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: When the oscillator begins at its maximum displacement of 0.3m at t = 0s, A = 0.3 and theta_0 = 0. This is found by knowing that the peak amplitude of the movement is A. Knowing that A = 0.3, we evaluate the equation at t = 0, and find a value of theta_0 that yields an initial displacement of 0.3m. When the oscillator reaches a maximum displacement of 0.3m, but the displacement at t = 0s is 0.15m, A = 0.3 and theta_0 = pi/3. Again, we begin with A = 0.3, as the maximum displacement. Knowing this, we evaluate at t = 0s, and find that theta_0 = pi/3 to yield an initial displacement of 0.15m. When the oscillator has a max velocity of 2 m/s, and is at its maximum displacement of 0.3m at t = 0s, A = 0.3 and theta_0 = 0. We begin with A = 0.3, and rationalize that if the maximum displacement is at t = 0, then theta_0 = 0, for the same reason that theta_0 = 0 in the first part. We can find the velocity equation by taking the derivative of the position function. Doing so, we realize that the maximum velocity is 6 m/s, and not 2m/s. If the oscillator has a maximum velocity of 2m/s, occurring at t = 0s, then A = 0.1m, and theta_0 = 3pi/2 + npi, where n = even integers. We begin with the derivative of our displacement equation, which is -Aωsin(ωt+theta_0). We already know that ω = 20, and our maximum velocity is 2m/s. We know that the maximum velocity occurs when the sine function portion is equal to -1 in this instance. This occurs when theta_0 is 3pi/2, and in increments of 360 deg (2pi). We then can determine if ω = 20, then A = 0.1m. Confidence rating: 2
.............................................
Given Solution: As seen in the preceding problem, a general solution to the equation is x = A cos(omega * t + theta_0), where omega = sqrt(k / m). For the current equation 5 x '' = -2000 x, this gives us omega = 20. In the current context omega = 20 radians / second. So x(t) = A cos( 20 rad / sec * t + theta_0 ). Maximum displacement occurs at critical values of t, values at which x ' (t) = 0. Taking the derivative of x(t) we obtain x ' (t) = - 20 rad / sec * A sin( 20 rad/sec * t + theta_0). The sine function is zero when its argument is an integer multiple of pi, i.e., when 20 rad/sec * t + theta_0 = n * pi, where n = 0, +-1, +-2, ... . A second-derivative test shows that whenever n is an even number, our x(t) function has a negative second derivative and therefore a maximum value. We can therefore pick any even number n and we will get a solution. If maximum displacement occurs at t = 0 then we have 20 rad / sec * 0 + theta_0 = n * pi so that theta_0 = n * pi, where n can be any positive or negative even number. We are free to choose any such value of n, so we make the simplest choice, n = 0. This results in theta_0 = 0. Now if x = .3 when t = 0 we have A cos(omega * 0 + theta_0) = .3 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think I did pretty well on this portion of this assignment. My trig skills are a bit rusty, but I think I reasoned things through pretty well and used my knowledge of calculus to help out. Im not sure about the third part. I spent awhile thinking about this, and just cant come to a solution where the maximum displacement is 0.3m, and the maximum velocity is 2m/s.
.............................................
Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Part II: Solutions of equations requiring only direct integration. `q006. Find the general solution of the equation x ' = 2 t + 4, and find the particular solution of this equation if we know that x ( 0 ) = 3. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x_g(t) = t^2+4t + c x_p(t) = t^2+4t+3 Confidence rating: 3
.............................................
Given Solution: Integrating both sides we obtain x(t) = t^2 + 4 t + c, where c is an arbitrary constant. The condition x(0) = 3 becomes x(0) = 0^2 + 4 * 0 + c = 3, so that c = 3 and our particular solution is x(t) = t^2 + 4 t + 3. We check our solution. Substituting x(t) = t^2 + 4 t + 3 back into the original equation: (t^2 + 4 t + 3) ' = 2 t + 4 yields 2 t + 4 = 2 t + 4, verifying the general solution. The particular solution satisfies x(0) = 3: x(0) = 0^2 + 4 * 0 + 3 = 3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q007. Find the general solution of the equation x ' ' = 2 t - .5, and find the particular solution of this equation if we know that x ( 0 ) = 1, while x ' ( 0 ) = 7. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x_g(t) = t^2-0.5t+C x_p(t) = t^2-0.5t+7 x_g(t) = t^3/3-t^2/4+7t+D x_p(t) = t^3/3-t^2/4+7t+1 These are all done with simple integration and calculus, easily done mentally. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: Integrating both sides we obtain x ' = t^2 - .5 t + c_1, where c_1 is an arbitrary constant. Integrating this equation we obtain x = t^3 / 3 - .25 t^2 + c_1 * t + c_2, where c_2 is an arbitrary constant. Our general solution is thus x(t) = t^3 / 3 - .25 t^2 + c_1 * t + c_2. The condition x(0) = 1 becomes x(0) = 0^3 / 3 - .25 * 0^2 + c_1 * 0 + c_2 = 1 so that c_2 = 1. x ' (t) = t^2 - .5 t + c_1, so our second condition x ' (0) = 7 becomes x ' (0) = 0^2 - .5 * 0 + c_1 = 7 so that c_1 = 7. For these values of c_1 and c_2, our general solution x(t) = t^3 / 3 - .25 t^2 + c_1 * t + c_2 becomes the particular solution x(t) = t^3 / 3 - .25 t^2 + 7 t + 1. You should check to be sure this solution satisfies both the given equation and the initial conditions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q008. Use the particular solution from the preceding problem to find x and x ' when t = 3. Interpret your results if x(t) represents the position of an object at clock time t, assuming SI units. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x(t) = t^3/3 - t^2/4 + 7t +1 x(3) = 27/3 - 9/4 + 21 + 1 = 115/4 m = 28.75m x(t) = t^2-t/2+7 x(3) = 9 - 3/2 + 7 = 29/2 m/s = 14.5 m/s Object is moving at 14.5m/s at 28.75m, when the clock reads 3s. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: Our solution was x(t) = t^3 / 3 - .25 t^2 + 7 t + 1. Thus x ' (t) = t^2 - .5 t + 7. When t = 3 we obtain x(3) = 3^3 / 3 - .25 * 3^2 + 7 * 3 + 1 = 28.75 and x ' (3) = 3^2 - .5 * 3 + 7 = 14.5. A graph of x vs. t would therefore contain the point (3, 28.75), and the slope of the tangent line at that point would be 14.5. x(t) would represent the position of an object. x(3) = 28.75 represents an object whose position with respect to the origin is 28.75 meters when the clock reads 3 seconds. x ' (t) would represent the velocity of the object. x ' (3) = 14.5 indicates that the object is moving at 14.5 meters / second when the clock reads 3 seconds. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): oK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q009. The equation x '' = -F_frict / m - c / m * x ', where the derivative is understood to be with respect to t, is of at least one of the forms listed below. Which form(s) are appropriate to the equation? x '' = f(x, x') x '' = f(t) x '' = f(x, t) x '' = f(x', t) x '' = f(x, x ' t) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: All but f(t) and f(x,t). Any form that includes x as a function is appropriate. Confidence rating: 3
.............................................
Given Solution: The right-hand side of the equation includes the function x ' but does not include the variable t or the function x. So the right-hand side can be represented by any function which includes among its variables x '. That function may also include x and/or t as a variable. The forms f(t) and f(x, t) fail to include x ', so cannot be used to represent this equation. All the other forms do include x ' as a variable, and may therefore be used to represent the equation. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q010. If F_frict is zero, then the function x in the equation x '' = -F_frict / m - c / m * x ' represents the position of an object of mass m, on which the net force is - c * x '. Explain why the expression for the net force is -c * x '. Explain what happens to the net force as the object speeds up. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: F = ma >> F_net = m*x x = F_net/m By matching up terms, we can conclude that -F_frict- c*x is the F_net of our equation. This means that when F_frict is 0, then our F_net is simply equal to -cx. As the object speeds up, the net force is going to increase. Confidence rating: 3 Given Solution: Newton's Second Law gives us the general equation m x '' = F_net so that x '' = F_net / m. It follows that x '' = -F_frict / m - c / m * x ' represents an object on which the net force is -F_frict - c x '. If F_frict = 0, then it follows that the net force is -c x '. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q011. We continue the preceding problem. If w(t) = x '(t), then what is w ' (t)? If x '' = - b / m * x ', then if we let w = x ', what is our equation in terms of the function w? Is it possible to integrate both sides of the resulting equation? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: w(t) = x(t) w(t) = -bw(t)/m dw/dt = -bw(t)/m dw/w = -bdt/m integ(1/w,dw) = integ(-b/m,dt) ln(w(t)) = -b*t/m w(t) = e^(-bt/m) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: If w(t) = x ' (t) then w ' (t) = (x ' (t) ) ' = x '' ( t ). If x '' = - b / m * x ', then if w = x ' it follows that x '' = w ', so our equation becomes w ' (t) = - b / m * w (t) The derivative is with respect to t, so if we wish to integrate both sides we will get w(t) = integral ( - b / m * w(t) dt), The variable of integration is t, and we don't know enough about the function w(t) to perform the integration on the right-hand side. [ Optional Preview: There is a way around this, which provides a preview of a technique we will study soon. It isn't too hard to understand so here's a preview: w ' (t) means dw / dt, where w is understood to be a function of t. So our equation is dw/dt = -b / m * w. It turns out that in this context we can sort of treat dw and dt as algebraic quantities, so we can rearrange this equation to read dw / w = -b / m * dt. Integrating both sides we get integral (dw / w) = -b / m integral( dt ) so that ln | w | = -b / m * t + c. In exponential form this is w = e^(-b / m * t + c). There's more, but this is enough for now ... ]. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Part III: Direction fields and approximate solutions `q012. Consider the equation x ' = (2 x - .5) * (t + 1). Suppose that x = .3 when t = .2. If a solution curve passes through (t, x) = (.2, .3), then what is its slope at that point? What is the equation of its tangent line at this point? If we move along the tangent line from this point to the t = .4 point on the line, what will be the x coordinate of our new point? If a solution curve passes through this new point, then what will be the slope at the this point, and what will be the equation of the new tangent line? If we move along the new tangent line from this point to the t = .6 point, what will be the x coordinate of our new point? Is is possible that both points lie on the same solution curve? If not, does each tangent line lie above or below the solution curve, and how much error do you estimate in the t = .4 and t = .5 values you found? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: At the point (0.2,0.3) the slope is 0.12 (substitute t = 0.2 and x = 0.3 into x). The equation of the tangent line at this point is (x-0.3) = 0.12(t-0.2). Moving along the tangent line to the t = 0.4 point, the x coordinate is 0.324. The slope at this point will be 0.2072. The new tangent line will be (x-0.324)=0.2072(t-0.4). Moving along the new tangent line to t = 0.6, the x coordinate is 0.36544. Since we are assuming that the slope is the same for the changes of t, we can probably safely conclude that the points lie below the solution curve. Confidence rating: 3
.............................................
Given Solution: At the point (.2, .3) in the (t, x) plane, our value of x ' is x ' = (2 * .3 - .5) * (.2 + 1) = .12, approximately. This therefore is the slope of any solution curve which passes through the point (.2, .3). The equation of the tangent plane is therefore x - .3 = .12 * (t - .2) so that x = .12 t - .24. If we move from the t = .2 point to the t = .4 point our t coordinate changes by `dt = .2, so that our x coordinate changes by `dx = (slope * `dt) = .12 * .2 = .024. Our new x coordinate will therefore be .3 + .024 = .324. This gives us the new point (.4, .324). At this point we have x ' = (2 * .324 - .5) * (.4 + 1) = .148 * 1.4 = .207. If we move to the t = .6 point our change in t is `dt = .2. At slope .207 this would imply a change in x of `dx = slope * `dt = .207 * .2 = .041. Our new x coordinate will therefore be .324 + .041 = .365. Our t = .6 point is therefore (.6, .365). From our two calculated slopes, the second of which is significantly greater than the first, it appears that in this region of the x-t plane, as we move to the right the slope of our solution curve in fact increases. Our estimates were based on the assumption that the slope remains constant over each t interval. We conclude that our estimates of the changes in x are probably a somewhat low, so that our calculated points lie a little below the solution curve. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q013. Consider once more the equation x ' = (2 x - .5) * (t + 1). Note on notation: The points on the grid (0, 0), (0, 1/4), (0, 1/2), (0, 3/4), (0, 1) (1/4, 0), (1/4, 1/4), (1/4, 1/2), (1/4, 3/4), (1/4, 1) (1/2, 0), (1/2, 1/4), (1/2, 1/2), (1/2, 3/4), (1/2, 1) (3/4, 0), (3/4, 1/4), (3/4, 1/2), (3/4, 3/4), (3/4, 1) (1, 0), (1, 1/4), (1, 1/2), (1, 3/4), (1, 1) can be specified succinctly in set notation as { (t, x) | t = 0, 1/4, ..., 1, x = 0, 1/4, ..., 1}. ( A more standard notation would be { (i / 4, j / 4) | 0 <= i <= 4, 0 <= j <= 4 } ) Find the value of x ' at every point of this grid and sketch the corresponding direction field. To get you started the values corresponding to the first, second and last rows of the grid are -.5, -.625, -.75, -.875, -1 0, 0, 0, 0, 0 ... ... 1.5, 1.875, 2.25, 2.625, 3 So you will only need to calculate the values for the third and fourth rows of the grid. List your values of x ' at the five points (0, 0), (1/4, 1/4), (1/2, 1/2), (3/4, 3/4) and (1, 1). Sketch the curve which passes through the point (t, x) = (.2, .3). Describe your curve. Is it increasing or decreasing, and is it doing so at an increasing or decreasing rate? According to your curve, what will be the value of x when t = 1? Sketch the curve which passes through the point (t, x) = (.5, .7). According to your curve, what will be the value of x when t = 1? Describe your curve and compare it with the curve you sketched through the point (.2, .3). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Substitution into x yields m = {-0.5, 0, 0.75, 1.75, 3 | (i,j) when i=j, i = 0,0.25,0.5,0.75,1} The curve is decreasing at an increasing rate from left to right when it passes through (0.2,0.3). It appears to have a horizontal asymptote at x = 0.25. According to the curve, the value of x when t = 1 will be 0.28. The curve is increasing at an increasing rate from left to right when it passes through (0.5, 0.7). It appears to have a horizontal asymptote at x = 0.25. Confidence rating: 2
.............................................
Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q014. We're not yet done with the equation x ' = (2 x - .5) * (t + 1). x ' is the derivative of the x(t) function with respect to t, so this equation can be written as dx / dt = (2 x - .5) * (t + 1). Now, dx and dt are not algebraic quantities, so we can't multiply or divide both sides by dt or by dx. However let's pretend that they are algebraic quantities, and that we can. Note that dx is a single quantity, as is dt, and we can't divide the d's. Rearrange the equation so that expressions involving x are all on the left-hand side and expressions involving t all on the right-hand side. Put an integral sign in front of both sides. Do the integrals. Remember that an integration constant is involved. Solve the resulting equation for x to obtain your general solution. Evaluate the integration constant assuming that x(.2) = .3. Write out the resulting particular solution. Sketch the graph of this function for 0 <= t <= 1. Describe your graph. How does the value of your x(t) function at t = 1 compare to the value your predicted based on your previous sketch? How do your values of x(t) at t = .4 and t = .6 compare with the values you estimated previously? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: dx / dt = (2 x - .5) * (t + 1) >> integ(1/(2x-0.5) dx) = integ(t+1 dt) >> ½ln(x-0.25)+c_1= t^2/2+t+c_2 >> x = e^(t^2+2t+C)+0.25 0.3 = e^(0.2^2+2(0.2)+C)+0.25 >> 0.05 = e^(0.2^2+0.4+C) >> 0.05 = Ce^(0.44) >> C = 0.05/e^(0.44) ≈ 0.0322 The particular solution is x = 0.0322e^(t^2+2t)+0.25. On the interval of 0<=t<=1, the solution appears to begin at approx. 0.2822, and slowly increases in an increasing fashion throughout the interval. My predictions were off quite a bit. Confidence rating: 3
.............................................
Given Solution: The equation is easily rearranged into the form dx / (2 x - .5) = (t + 1) dt. Integrating the left-hand side we obtain 1/2 ln | 2 x - .5 | Integrating the right-hand side we obtain t^2 / 2 + 4 t + c, where the integration constant c is regarded as a combination of the integration constants from the two sides. Thus our equation becomes 1/2 ln | 2 x - 5 | = t^2 / 2 + t + c. Multiplying both sides by 2, then taking the exponential function of both sides we get exp( ln | 2 x - 5 | ) = exp( t^2 + 2 t + c ), where as before c is an arbitrary constant. Since the exponential and natural log are inverse functions the left-hand side becomes | 2 x + .5 |. The right-hand side can be written e^c * e^(t^2 + 8 t), where c is still an arbitrary constant. e^c can therefore be any positive number, and we replace e^c with A, understanding that A is a positive constant. Our equation becomes | 2 x - .5 | = A e^(t^2 + 2 t). For x > -.25, as is the case for our given value x = .3 when t = .2, we have 2 x - .5 = A e^(t^2 + 2 t) so that x = A e^(t^2 + 2 t) + .25. Using x = .3 and t = .2 we find the value of A: .05 = A e^(.2^2 + 2 * .2) so that A = .05 / e^(.44) = .03220, approx.. Our solution function is therefore x(t) = .05 / e^(.44) * e^(t^2 + 2 t) + .25, or approximately x(t) = .03220 e^(t^2 + 2 t) + .25 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: `q015. OK, this time we are really going to be done with this equation. Again, x ' = (2 x - .5) * (t + 1) Along what line or curve is x ' = 1? Along what line or curve is x ' = 0? Along what line or curve is x ' = 2? Along what line or curve is x ' = -1? Sketch these three lines and/or curves for 0 <= t <= 1. Along each of these lines x ' is constant. Along each sketch 'slope segments' with slopes equal to the corresponding value of x '. How consistent is your sketch with your previous sketch of the direction field? Sketch a solution curve through the point (.2, .25), and estimate the coordinates of the t = 1 point on this curve. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1 = (2x-0.5)(t+1) >> solve for x >> x = (1/(t+1)+0.5)/2 >> x = 1/(2t+2)+0.25 0 = (2x-0.5)(t+1) >> solve for x >> x = 0.25 2 = (2x-0.5)(t+1) >> solve for x >> x = (2/(t+1)+0.5)/2 >> x = 1/(t+1) + 0.25 -1 = (2x-0.5)(t+1) >> solve for x >> x = (-1/(t+1) + 0.5)/2 >> x = -1/(2t+2)+0.25 I would say my sketch is pretty consistent with my previous sketch. The coordinates of the t = 1 point on this curve is (1, 0.25). The curve is a horizontal line. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: x ' = 1 when (2 x - .5) * (t + 1) = 1. Solving for x we obtain x = 1/2 ( 1 / (t + 1) + .5) = 1 / (2(t + 1)) + .25. The resulting curve is just the familiar curve x = 1 / t, vertically compressed by factor 2 then shifted -1 unit in the horizontal and .25 unit in the vertical direction, so its asymptotes are the lines t = -1 and x = .25. The t = 0 and t = 1 points are (0, .75) and (1, .5). Similarly we find the curves corresponding to the other values of x ': For x ' = 0 we get the horizontal line x = .25. Note that this line is the horizontal asymptote to the curve obtained in the preceding step. For x ' = 2 we get the curve 1 / (t + 1) + .25, a curve with asymptotes at t = -1 and x = .25, including points (0, 1.25) and (1, .75). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating: OK These questions can be challenging if you are rusty on first-year calculus and trigonometric functions. Fortunately we won't be using these functions a lot near the beginning of the course, but they become very important later, and if you're rusty, or if you never really mastered these functions, you want to get a good start now. If you are rusty, try to answer the following without the use of a calculator: Quick refresher on trigonometric functions: What are the maximum and minimum possible values of cos(theta)? max: 1; min: -1 #$&* For what values of theta does cos(theta) take its maximum value, and for what values of theta is cos(theta) equal to zero? max: 0, 2pi kpi for all k = even integer values; zero: pi/2, 3pi/2 xpi/2 for all x = odd integer values #$&* For what values of t does y = cos(5 t) take its maximum value? For what t values does this function take its minimum value? max: t=0, 2pi/5 npi/5 for all n = even integer values; min: t=pi/5 npi/5 for all n = odd integer values #$&* What is the maximum possible value of y = 4 cos(83 t)? max = 4 #$&* Find at least one value of t for which y = cos(2 t + pi) takes the value zero. y = 0 @ t = pi/4 #$&* Find at least one value of t for which y = 3 cos(2 t + pi/4) takes its maximum value. y = 3 @ t = 7pi/8 #$&* Find at least one value of t for which the derivative of y = cos(t) is zero. y = 0 @ t = 0 #$&* Find at least one value of t for which the derivative of y = cos(t) is zero, and the second derivative is negative. y = -sin(t) = 0 @ t = 0 y = -cos(t) = -1 @ t = 0 #$&* Find the critical points of y = 4 sin(2 pi t + pi/2), then using a second-derivative test find the values of t at which this function takes its maximum and minimum values. y= 8pi*cos(2pi*t+pi/2) = -8pi*sin(2pi*t) = 0 @ t = 0, 0.5, 1, 1.5 y = -16pi^2*sin(2pi*t+pi/2) = -16pi^2 *cos(2pi*t). Since we know that the cos function takes a max value at 0 and 2pi: maximum values occur at t = all real integers (although if t signifies time, obviously we cant have negative time, so in that case it would be all positive real integers). We know that the cos function takes a min value at pi, 3pi, 5pi, etc, then we know that minimum values occur at t = ½, 3/2, 5/2, n/2 for all odd integers. Again, if t is in a unit of time, then it limits our range to positive odd integers. Feel free to submit a copy of these questions with your answers. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q015. OK, this time we are really going to be done with this equation. Again, x ' = (2 x - .5) * (t + 1) Along what line or curve is x ' = 1? Along what line or curve is x ' = 0? Along what line or curve is x ' = 2? Along what line or curve is x ' = -1? Sketch these three lines and/or curves for 0 <= t <= 1. Along each of these lines x ' is constant. Along each sketch 'slope segments' with slopes equal to the corresponding value of x '. How consistent is your sketch with your previous sketch of the direction field? Sketch a solution curve through the point (.2, .25), and estimate the coordinates of the t = 1 point on this curve. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 1 = (2x-0.5)(t+1) >> solve for x >> x = (1/(t+1)+0.5)/2 >> x = 1/(2t+2)+0.25 0 = (2x-0.5)(t+1) >> solve for x >> x = 0.25 2 = (2x-0.5)(t+1) >> solve for x >> x = (2/(t+1)+0.5)/2 >> x = 1/(t+1) + 0.25 -1 = (2x-0.5)(t+1) >> solve for x >> x = (-1/(t+1) + 0.5)/2 >> x = -1/(2t+2)+0.25 I would say my sketch is pretty consistent with my previous sketch. The coordinates of the t = 1 point on this curve is (1, 0.25). The curve is a horizontal line. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.............................................
Given Solution: x ' = 1 when (2 x - .5) * (t + 1) = 1. Solving for x we obtain x = 1/2 ( 1 / (t + 1) + .5) = 1 / (2(t + 1)) + .25. The resulting curve is just the familiar curve x = 1 / t, vertically compressed by factor 2 then shifted -1 unit in the horizontal and .25 unit in the vertical direction, so its asymptotes are the lines t = -1 and x = .25. The t = 0 and t = 1 points are (0, .75) and (1, .5). Similarly we find the curves corresponding to the other values of x ': For x ' = 0 we get the horizontal line x = .25. Note that this line is the horizontal asymptote to the curve obtained in the preceding step. For x ' = 2 we get the curve 1 / (t + 1) + .25, a curve with asymptotes at t = -1 and x = .25, including points (0, 1.25) and (1, .75). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK Self-critique rating: OK These questions can be challenging if you are rusty on first-year calculus and trigonometric functions. Fortunately we won't be using these functions a lot near the beginning of the course, but they become very important later, and if you're rusty, or if you never really mastered these functions, you want to get a good start now. If you are rusty, try to answer the following without the use of a calculator: Quick refresher on trigonometric functions: What are the maximum and minimum possible values of cos(theta)? max: 1; min: -1 #$&* For what values of theta does cos(theta) take its maximum value, and for what values of theta is cos(theta) equal to zero? max: 0, 2pi kpi for all k = even integer values; zero: pi/2, 3pi/2 xpi/2 for all x = odd integer values #$&* For what values of t does y = cos(5 t) take its maximum value? For what t values does this function take its minimum value? max: t=0, 2pi/5 npi/5 for all n = even integer values; min: t=pi/5 npi/5 for all n = odd integer values #$&* What is the maximum possible value of y = 4 cos(83 t)? max = 4 #$&* Find at least one value of t for which y = cos(2 t + pi) takes the value zero. y = 0 @ t = pi/4 #$&* Find at least one value of t for which y = 3 cos(2 t + pi/4) takes its maximum value. y = 3 @ t = 7pi/8 #$&* Find at least one value of t for which the derivative of y = cos(t) is zero. y = 0 @ t = 0 #$&* Find at least one value of t for which the derivative of y = cos(t) is zero, and the second derivative is negative. y = -sin(t) = 0 @ t = 0 y = -cos(t) = -1 @ t = 0 #$&* Find the critical points of y = 4 sin(2 pi t + pi/2), then using a second-derivative test find the values of t at which this function takes its maximum and minimum values. y= 8pi*cos(2pi*t+pi/2) = -8pi*sin(2pi*t) = 0 @ t = 0, 0.5, 1, 1.5 y = -16pi^2*sin(2pi*t+pi/2) = -16pi^2 *cos(2pi*t). Since we know that the cos function takes a max value at 0 and 2pi: maximum values occur at t = all real integers (although if t signifies time, obviously we cant have negative time, so in that case it would be all positive real integers). We know that the cos function takes a min value at pi, 3pi, 5pi, etc, then we know that minimum values occur at t = ½, 3/2, 5/2, n/2 for all odd integers. Again, if t is in a unit of time, then it limits our range to positive odd integers. Feel free to submit a copy of these questions with your answers. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!