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First order differential equations.

  1. Apr 17, 2012 #1
    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 [tex]y'=x+y , y(0)=1, h=0.1[/tex] I can apply the Euler's method:
    [tex]y_1=y_0+hf(x_0,y_0)=1+0.1(0+1)=1+0.1=1.1[/tex]

    But the following equations all contain the term y''.
    [tex]y''-0.2(1-y^2)y'+y=0 , y(0)=0.1, y'(0)=0.1, calculate: y(0.2)[/tex]
    [tex]-x^2y''-2xy'+2y=-4x^2 , y(1)=0, y'(1)=1, calculate: y(1.1)[/tex]
    [tex]y''=2y'/x-2y/x^2-1/x^2 , y(1)=0, y'(1)=1, calculate: y(1.1)[/tex]
    How am I supposed to turn them into first order DEs please?
     
  2. jcsd
  3. Apr 17, 2012 #2
    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).
     
  4. Apr 17, 2012 #3
    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?
     
  5. Apr 17, 2012 #4
    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.
     
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