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Homework Help: Linear Equations

  1. Aug 6, 2006 #1
    Hi, I am finding some of this confusing, can someone explain this?

    so I undersand that [tex]xy' + y = (xy)'[/tex]

    lets say that I have [tex]2ty' + 4y = 2t^3[/tex], what is (xy)'?

    would it just become [tex]d/dx(2ty) = 2t^3[/tex]?
     
    Last edited: Aug 6, 2006
  2. jcsd
  3. Aug 6, 2006 #2

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    You're mixing up x and t. If you just want to solve:

    [tex]2t \frac{dy}{dt}+4y=2t^3[/tex]

    then you should use integrating factors. That is, find a pair of functions f(t) and g(t) such that (substituting back y' for dy/dt):

    [tex]f(t) (2t y'+4y)= \frac{d}{dt} (g(t) y)[/tex]

    Right away you can see that 2tf(t)=g(t), and then you can get a simple ODE to solve for g(t). Now you multiply across in the original equation:

    [tex]f(t) (2t y' +4y)=\frac{d}{dt} (g(t) y)=f(t) 2t^3[/tex]

    and then you just need to integrate. Note that the case (xy)' you describe first is another example of integrating factors, in that case with g(x)=x. In this case, g will be different.
     
    Last edited: Aug 6, 2006
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