First order differential equations.

[bubokribuck]

Hi, I have problems rewriting equations with the term y'' as a system of first order differential equations.

I've been given several equations and was told to write them as 1st order DEs, then calculate the numerical solution using the Euler's modified method.

I know that y'=f(x,y), so if $$y'=x+y , y(0)=1, h=0.1$$ I can apply the Euler's method:
$$y_1=y_0+hf(x_0,y_0)=1+0.1(0+1)=1+0.1=1.1$$

But the following equations all contain the term y''.
$$y''-0.2(1-y^2)y'+y=0 , y(0)=0.1, y'(0)=0.1, calculate: y(0.2)$$
$$-x^2y''-2xy'+2y=-4x^2 , y(1)=0, y'(1)=1, calculate: y(1.1)$$
$$y''=2y'/x-2y/x^2-1/x^2 , y(1)=0, y'(1)=1, calculate: y(1.1)$$
How am I supposed to turn them into first order DEs please?

 

[sunjin09]

Instead of writing y''=f(x,y), write z(t)=y'(t), so that y''(t)=z'(t), solve for y(t) and z(t).

 

[bubokribuck]

Hi sunjin09, thanks for your reply, but there are two variables x and y involved though, how can I convert (x,y) into (t) please?

 

[sunjin09]

Oh right, your free variable is x, not t, so, you should work with the functions y(x) and z(x)=y'(x), not t.