Answer to Question #216241 in Differential Equations for Wavie

Question

Use Euler’s method with h = 0.25 to obtain a numerical solution of

dy/

dx= −xy2

subject to y(0) = 2, giving approximate values of y for 0 $\leq$ x $\geq$ 1. Work throughout

to three decimal places and determine the exact solution for comparison.

Answer 1

Solution

Exact solution of DE: dy/y² = -xdx  =>   -1/y = -x²/2-C  =>  y = 2/(x²+2C)

y(0) = 2   => 2 = 2/(0+2C)  =>  C = ½   =>  y(x) = 2/( x²+1)

For the first-order differential equation with the initial value

y’(x) = F(x,y y(x)),  y(x₀) = y₀

according to the Euler method

y_n+1 = y_n + h*F(x_n, y_n)

where h – step, x_n = h*n, y_n = y(x_n), n = 0,1,2…N

In this case F(x,y) = -xy² , h = 0.25 , x₀ = 0 , y₀ = 2.

Calculations gives such results (last column is the exact solution):

n            x_n           y_n           y(x_n)

0            0            2             2

1            0.25      2             1.882

2            0.5         1.75       1.6

3            0.75      1.367    1.28

4            1            1.017    1