Exercises

01

Euler’s method struggles with large slopes

Consider the IVP:

$$y^{\prime }=\frac{3t^2}{3y^2-4},y(1)=0.$$

Desmos: https://www.desmos.com/calculator/gbijdaprd3

• Use Euler’s method to compute an approximate solution at the times $t=1.2,1.4,1.6,1.8$ using $h=0.10$.
• Use Euler’s method to compute an approximate solution at the times $t=1.2,1.4,1.6,1.8$ using $h=0.05$.
• By considering the slope field, possibly in conjunction with an exact solution, can you explain why the approximate solutions differ so much at $t=1.8$?

02

Euler’s method becomes the left-endpoint rule

Show that if the ODE is amenable to solution by direct integration, meaning $f(t,y)$ actually does not depend on $y$, then Euler’s method amounts to the left-endpoint rule for integration of $f(t)$.

03

The Runge-Kutta method becomes the Simpson rule

Show that if the ODE is amenable to direct integration, so $f(t,y)$ actually does not depend on $y$, then the Runge Kutta method amounts to the Simpson rule for integrating $f(t)$.

Recall that the Simpson rule comes from fitting parabolas. It is given by:

$${\int }_a^{a+h}f(x)dx=\frac{h}{6}(f(a)+4f(a+h/2)+f(a+h)).$$

04

Three-step Adams method

Compute the formula for the three-step Adams method.