Introductory Differential Equations using Sage - William Stein

1.7. NUMERICAL SOLUTIONS II - RUNGE-KUTTA AND OTHER METHODS 43
The plot of the approximation to x(t) is given in Figure 1.14.
Figure 1.14: Euler’s method with h = 1/3 for x ′′ − 3x ′ + 2x = 1, x(0) = 0, x ′ (0) = 1.
Exercise: Use Sage and Euler’s method with h = 1/3 for the following problems:
1. (a) Use Euler’s method to estimate x(1) if x(0) = 1 and dx
dt = x + t2 , using 1,2, and
4 steps.
(b) Find the exact value of x(1) by solving the ODE (it is a linear ODE).
2. Find the approximate values of x(1) and y(1) where
{ x ′ = x + y + t, x(0) = 0,
y ′ = x − y, y(0) = 0,
3. Find the approximate value of x(1) where x ′ = x 2 + t 2 , x(0) = 1.
1.7 Numerical solutions II - Runge-Kutta and other methods
The methods of 1.6 are sufficient for computing the solutions of many problems, but often
we are given difficult cases that require more sophisticated methods. One class of methods
are called the Runge-Kutta methods, particularly the fourth-order method of that class
since it achieves a popular balance of efficiency and simplicity. Another class, the multistep
methods, use information from some number m of previous steps. Within that class, we
will briefly describe the Adams-Bashforth method. Finally, we will say a little bit about
adaptive step sizes - i.e. changing h adaptively depending on some local estimate of the
error.