Assignment 4

due Friday, Feb. 8

Note: An entry such as ``2.1.6'' means question 6 in section 2.1 of Boyce and DiPrima. You will find the Differential Equations Calculator applet useful for doing the numerical calculations.

8.3.11 Solution

8.3.14 Solution

2.6.8 Solution

E.1      A student is using a numerical method to approximate $y(3)$ for an initial value problem $$y' = f(x,y)$$, $y(0) = 1$. Using step size $h = 0.2$ she gets an answer of $1.752$, while with $h=0.1$ she gets $1.692$. The true answer happens to be $1.671$. Which method do you think she is using: Euler, Improved Euler, Runge-Kutta, or some other method? Hint: what does $p$ appear to be for this method?   Solution

E.2      Use the Fourth-Order Runge-Kutta method to find an approximate value for $y(3)$ where $y$ is the solution of the initial value problem $$y' = 2 x y - x^2$$, $y(1)=1$. Use Richardson extrapolation to estimate a step size for which the error should be less than $$10^{-6}$$, and use Richardson extrapolation again to check this.

Note: Don't try to solve this equation exactly; instead, use Richardson extrapolation to estimate the error. To be on the safe side, it's best to use a step size slightly smaller than the one for which you would predict an error of $$10^{-6}$$.  Solution

E.3      Find (approximately) the largest step size $h$ such that, when the Fourth Order Runge Kutta method is used on the differential equation $y' = -100 y$, $\vert y_1\vert \le \vert y_0\vert$ (and thus the method is stable for this equation).   Solution