As always, rework my class examples SUCCESSFULLY ON YOUR OWN before attempting the homework.

Preliminary Questions:

1. How many solutions does an ordinary differential equation (ODE) generally have?
2. What point is described by the initial condition y(4) = –7?
3. Suppose we have ODE  y′ = y cos x. What is the slope of the solution at point (π, 5)?
   As a follow-up to this, what is f (x_n , y_n )?
   What is f (x_n+1 , y_n+1 )?
4. Suppose we have ODE  y′ = x cos y. What is the slope of the solution at point (π, 5)?
   As a follow-up to this, what is f (x_n , y_n )?
   What is f (x_n+1 , y_n+1 )?

This is a running assignment. Work through these problems AS we cover each method.

PROBLEM:  Approximate the solution of the initial value problem (IVP)

y'   =   5 sin(2x)   –   y ,

y(0)   =   –1 ,

from x = 0 to x = 1 with stepsize h = 0.2 using:

1. Euler's method,

2. the implicit Euler method,

3. the trapezoidal method,

4. the modified Euler (predictor-corrector) method, (Make sure you can do this method! Hint hint!)

5. the classical Runge-Kutta method. (Compute y₁ and y₂ only.)

For comparison, from MATH-204 we find that the exact solution of the IVP is

y*(x)   =   e^–x + sin(2x) – 2 cos(2x) .

In each method above, determine the error of the approximation at each node. For example, the error at node n is

E_n   =   y*(x_n) – y_n .

Here are the ANSWERS you should obtain.

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