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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.

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.

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 ) .

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