# First order differential equations.

1. Apr 17, 2012

### 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?

2. Apr 17, 2012

### 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).

3. Apr 17, 2012

### 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?

4. Apr 17, 2012

### 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.