mms://media.vhcc.edu/dsmith/differential_equations/introduction/
A_slope_rule_is_a_differential_equation
Use_the_slope_rule_to_sketch_approximate_curves
Using the rule we can construct an approximate curve starting at any point
Check_the solution_against_the approximate_curve
The rule_y ' = t / 2 was_a_simple_function_of_t and was therefore_easily_integrated
something_that_should_have_bothered_you
fundamental_theorem_of_calculus
integraing_two_simple_equations
modeling_ball_on_ramp_with_drag
modeling_with_linear_drag_force
pendulum_in_changing_water_depth
form_of_equation_pendulum_in_water
equation_pendulum_in_rising_water
forms_of_some_differential_equations
reference_circle_model_of_sine_and_cosine
reference_circle_model_of_sine_and_cosine_2
circular_model_of_sine_and_cosine_3
circular_model_of_sine_and_cosine_4
circular_model_of_sine_and_cosine_5
circular_model_of_sine_and_cosine_6
circular_model_of_sine_and_cosine_7
A rule which gives a slope for every point is a differential equation
If we have a rule for calculating the slope at any point in a region of the y vs. t plane, then we can use this rule to construct a curve passing through any point of the region.
We simply take the coordinates of our starting point and plug them into our rule to get the slope at that point. Then we sketch a short line segment through the point, with the segment having the slope given by the rule.
We then go to the end of our segment, determine its coordinates and plug the result into our rule. This gives us a new slope. We sketch another segment through our new point, giving the segment the new slope.
There's something that should have bothered you in the preceding.
The slope rule appears to give a different result with every new point, which is what we expect.
When we draw a straight line segment, however, we aren't allowing the slope to change. We don't use a new slope until we get to the end of our segment.
The point at the end of our segment will already diverge at least a little from the accurate curve. We then use this point, which is close but not quite right, to calculate the slope of our next segment, and we repeat the process. Our new point will again be a little off, both because we started from a point that isn't quite right, then because we used a straight line to approximate a curve that probably isn't straight.
So our with every new segment, our curve will tend to diverge more and more from the actual, ideal curve through our original starting point.
We can remedy this to an extent by sketching shorter segments, allowing us to recalculate the slope after a shorter interval. This will make our curve more accurate, though it won't completely solve the problem. The downside of using shorter segments is that it takes a lot longer to sketch our curve.
Another remedy is to sketch our segment, calculate the slope at the end of the segment, average this with our original slope and re-sketch our segment using the averaged slope. This results in a drastic improvement in the accuracy of our curve without having to resort to very short segments.
A slope rule is a differential equation
A rule which specifies the slope at every point of some region of the y vs. t plane is a differential equation.
Whatever the rule, it can be written in the form f(t, y), a function of the two variables y and t.
The slope is y ' = dy/dt.
So the differential equation is
dt / dt = f(t, y)
or if we prefer
y ' = f(t, y).
22:55 110531
We can use the slope rule to sketch approximate curves
Starting at any point within our region, we can sketch an approximate curve by following the slope field.
23:03 110531
Alternatively, we can start at any point, sketch a short segment, recalculate and repeat
23:10 110531
The equation y ' = t / 2
Check_the solution_against_the approximate_curve
The rule_y ' = t / 2 was_a_simple_function_of_t and was therefore_easily_integrated
If we integrate both sides of this equation we get
y = t^2 / 4 + c.
If we so choose we can emphasize that y is a function of t by writing our solution
y(t) = t^2 / 4 + c.
If we plug this expression back into our equation we get
(t^2 / 4 + c) ' = t / 2
Taking the derivative of the left-hand side we obtain the identity
t / 2 = t / 2,
verifying that our function is a solution of the equation.
We can check our solution against the given curve.
In this case our estimated solution curve appears to intercept the y axis at or near the point (0, 0.3).
If y(t) = t^2 / 4 + c takes value 0.3 and t = 0, then we conclude that our value of c for this curve must be 0.3.
Our estimated curve should therefore correspond fairly well with the function
y(t) = t^2 / 4 + 0.3.
For example, at t = 2 our function value should be about y(2) = 2^2 / 4 + .3 = 1.3.
Plotting the point (2, 1.3) on our graph, we find that our approximate curve comes reasonably close to this point.
Our rule was a simple function of t, therefore easily integrated.
Our equation was y ' = t / 2, which is of the form y ' = f(t, y) with f(t, y) = t/2.
This function is very easily integrated with respect to t.
So our equation was easily solved.
It's not always so.
In our first example, the rule was pretty simple:
y ' = y / 5 + t / 4.
We can easily integrate the left-hand side of this equation. y ' means the derivative of y with respect to t, so we integrate with respect to t.
The left-hand side will just be y.
The right-hand side will be the integral of y/5 + t/4 with respect to t.
We can easily integrate t / 4 with respect to t, obtaining t^2 / 2.
However we can't integrate y / 5 with respect to t. y stands for a function y(t), and we don't know that function. We can't integrate an as-yet-unknown function with respect to t.
So we're going to have to forget the idea of integrating y/5 with respect to t, and find some trick that allows us to figure out the function y(t).
Most of a differential equations course consists of finding tricks to solve various types of differential equations.
The trick for the equation y ' = y / 5 + t / 4 will come up soon, and it isn't all that difficult once you've seen it.
23:42 110531
The