#$&* course Phy 279 3/6 3 q_a_06________________________________________
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Given Solution: ` Self-critique: ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. In general air resistance has components which depend on v and on v^2, where v is the velocity of an object. If v ' = -5 t + 4 t v + t v^2, what is the solution of the equation for which v(0) = 10? Over what t interval can we expect this solution to exist? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: v’/(5+4v+v^2) - t = 0 Int(v’/(5+4v+v^2)), letting u= arctan(v + 2), du = 1/(1 + (v+2)^2) dv dv = (5+4v+v^2) du, plugging back into our equation Int((1/(5+4v+v^2))* 5+4v+v^2 du) = Int(1 du) = u = arctan(v + 2) ??????I’m not this is the correct approach but I’ve spent some time on this one and this is the best I can come with??????? v’/(5+4v+v^2) - t = 0, after integrals are taken arctan(v + 2) - t^2/2 = C arctan(v+2) = t^2/2 + c v+2 = tan( t^2/2 + c), finally v = -2 + tan( t^2/2 + c),solution is 10 = -2 + tan(c) c = arctan(8) = 1.466 v(t) = -2 + tan( t^2/2 + 1.466) tan fn is not defined at pi/2 + n*pi, where n is any integer ???I don’t believe this is correct, but I can’t see where my error has been?????
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Given Solution: ` Self-critique: ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q003. Show that the equation 2 y tan(t) dy + (y^2 tan(t) sec (t) + 1 / (2 sqrt(t) ) dt = 0 is of the form M dy + N dt = 0, with M_t = N_y. Integrate to find the function F(t, y) for which F_t = N, and F_y = M. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Already completed this confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ` Self-critique: ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. Consider the equation y ' + p(t) y = q(t) y^3. Let v(t) = y(t)^m, where the value of m has yet to be determined. What is dv/dt? What therefore is dy/dt in terms of v and dv/dt? What is y in terms of v? Rewrite the original equation in terms of p(t), q(t), v and dv/dt. Choose m so that the v in the right-hand side has power 0. Now the equation is linear nonhomogeneous order 1. `q What integrating factor allows us to solve the equation? NOTE: This is an example of a Bernoulli Equation, which is of the form y ' + p(t) y = q(t) y^n. This equation differs from a first-order linear homogeneous equation by the factor y^n on the right-hand side. The general method is to change the variable to v = y^m, then choose the value of m that makes the equation linear. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Already completed this confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ` Self-critique: ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q005. A population function P(t) satisfies dP/dt = k P, as long as space and resources are unlimited. That is, the rate of growth is proportional to the population. If, however, the population grows too much, it approaches the carrying capacity L of its environment, and its rate of growth becomes also proportional to (L - P). Thus the rate of population growth is jointly proportional to P and (L - P). This gives us the equation dP/dt = k P ( L - P ). This equation is separable. Find its general solution. You will need to use partial fractions. If L = 1000, P(0) = 100 and P(1) = 200, what is the value of k? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Already completed confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ` Self-critique: ------------------------------------------------ Self-critique Rating: "