Numerical Methods and Optimization

156 Numerical Methods and Optimization: An Introduction

Use Euler’s method with h =0.1 to approximate y(0.2).

Denoting by z = y



, the given problem is reduced to the following system:

y



= z

z



=(1− y

2

)z −y

y(0) = 0.2

z(0) = 0.2.

Applying Euler’s method to this system gives



y

k+1

z

k+1



=



y

k

z

k



+ h



z

k

(1 − y

2

k

)z

k

− y

k



,k ≥ 0.

Using y

0

= z

0

=0.2; h =0.1, we obtain



y

1

z

1



=



0.22

0.1992



.So,y

2

=

0.22 + 0.1 · 0.1992 = 0.23992.

Exercises

7.1. Consider the IVP y



= −xy, y(1) = 2.

(a) Verify that y(x)=2exp

$

1

2

(1 − x

2

)

%

is the solution.

(b) Apply Picard’s method three times.

7.2. Solve the equations

(i) y



= −

x

2

+exp(y)

x(1+exp(y))

,y(1) = 0;

(ii) y



=

y(1−x

2

y

2

)

x(1+x

2

y

2

)

,y(1) = 1

on the