Mathematics 132 – Calculus for Physical and Life Sciences 2 Discussion 6 – Slope Fields April 22, 2005

Your Group:

A) Which of the following slope fields goes with which differential equation?

Plot I:  Plot II:  Plot III:  Plot IV:  Plot V:  Plot VI:

B) You should have found that plot III corresponds to differential equation (e) in part A.

1) On the plot of III, sketch the solutions passing through the points (0, 1), (0, −1), (0, 0) (you should have three different graphs).

2) The exact formula for the solution of equation (e) or slope field III with y(0) = −1 is y(x) = x − 1. Verify that this function does solve equation (e):

Answer:

C) To get numerical approximations to values of the solution y = y(x) of a differential equation \frac{dy}{dx} = f(x, y) with given initial value y(x_{0}) = y_{0}, we could proceed as follows:

• Pick a “step size” h (like the Δx for a Riemann sum).

• Start at the point (x_{0}, y_{0}), “sample” the slope function there and go along a straight line of slope f(x_{0}, y_{0}) until you get to x_{1} = x_{0} + h. The y-coordinate on that line with x = x_{1} is y_{1} ≐ y(x_{1}).

• Now “sample” the slope function at (x_{1}, y_{1}) and go along a straight line of slope f(x_{1}, y_{1}) until you get to x_{2} = x_{1} + h = x_{0} + 2h. The y-coordinate on that line is y_{2} ≐ y(x_{2}).

• Repeat the process as often as needed (for instance, until x_{k} = some x value where you want to know an approximate solution value.

The process described above leads to the equation

y_{k} = y_{k−1} + f(x_{k−1}, y_{k−1})h

for producing y_{k} from x_{k−1}, y_{k−1}, h. For instance, if we are looking for numerical approximations to the solution of \frac{dy}{dx} = 2y with y(0) = 1, we might choose h = 0.2. Then (x_{0}, y_{0}) = (0, 1), x_{1} = 0.2, and

y_{1} = y_{0} + f(x_{0}, y_{0})h = 1 + 2(1)(.02) = 1.04

Continue this process to approximate the solution of \frac{dy}{dx} = 2y with y(0) = 1 at x_{2} = .4, x_{3} = .6, x_{4} = .8, and x_{5} = 1.0. Present your work as a table of values at these x’s.

k x_{k} y_{k}
0 0  1
1 .2 \underline{1.04}
2 .4
3 .6
4 .8
5 1.0

Assignment

Group writeups due no later than Monday, April 25.