Example Problem Set 9

(A) When numerically computing the solution to an ordinary differential equation (ODE) that involves higher-than first-order derivatives,

(i) What is to be done about the higher-than first-order terms, and how can this be accomplished?

(ii) Illustrate this step for the following ODE, in which functions r(x) and q(x)are known and we seek to compute the solution y(x):

$$\frac{d^{2}y}{dx^{2}} + q(x)\frac{dy}{dx} = r(x)$$

(B) (i) State the incrementing rule for the Euler method of numerical integration, in terms of:

• f(x_n), the estimate of the solution f(x) at the current point x_n
• f(x_n+1), the new estimate of f(x) for the next point x_n+1
• the integration stepsize h, which is the interval ( x_n+1 - x_n)
• $f^{\prime}(x_{n})$, the expression given by the ODE for the derivative of the desired solution f(x) at the current point x_n

(ii) What might happen to your solution if the stepsize h is too large?

(iii) What might happen to your solution if the stepsize h is too small?

(iv) What is the primary advantage of the fourth-order Runge-Kutta method over the Euler method for numerical integration of ODEs?