# Homework 9 # Problem 1

Given x 0 ( t ) = x

2

, x (0) = 1. Find the interval where the solution exists and determine whether or not the solution is unique. Compare the interval of existence with the one that corresponds to the exact solution.

# Problem 2

Derive the modified Euler’s method, x ( t + h ) = x ( t ) + h ( f ( t + 0 .

5 h ) , x ( t ) + 0 .

5 hf ( t, x ( t ))) by performing Richardson’s extrapolation on Euler’s method using step sizes h and h/ 2.

# Problem 3

λx

Show that when the fourth-order Runge-Kutta method is applied to the problem the formula for advancing this solution will be x 0 ( t ) = x ( t + h ) = [1 + hλ +

1

2 h

2

λ

2

+

1

6 h

3

λ

3

+

1

24 h

4

λ

4

] x ( t ) .

# Problem 4

Consider the following equation x 00 + 4 x = 0 (1) with x (0) = 0, x 0 (0) = 1. This equation describes the motion of an undamped 2 springmass system. The system possesses energy conservation property, i.e., the sum of kinetic

(0 .

5 x 0 ( t )

2

) and potential (2 x

2

) energies are constant

E ( t ) =

1

2 x ( t )

0

2

+ 2 x ( t )

2

=

1

2 x (0)

0

2

+ 2 x (0)

2

.

(2) a) Write a program for solving the equation using forward Euler method b) Write a program for solving the equation using fourth order Runge-Kutta method

Compare the numerical solution with the exact solution. Calculate the energy and compare with constant energy corresponding to the exact solution.

Use dt (= h ) = 0 .

1, N = 5000, where N is the total number of time steps.

1