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Math Help - [SOLVED] Differential Equations

  1. #1
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    Question [SOLVED] Differential Equations

    Hi there, I have a question on differential equations I dont understand, here it is:

    find the general solution:

    t (dx/dt) = x + (sin(x/t))^2

    I think you have to use substitution for this one but dont know how, any ideas?
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  2. #2
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    Define u=\frac{x}{t}. Then \frac{du}{dt}=-\frac{x}{t^2}+\frac{1}{t}\frac{dx}{dt}. Now, our original equation tells us \frac{dx}{dt}=\frac{x}{t}+\frac{1}{t}\sin^2\frac{x  }{t}, and so:
    \frac{du}{dt}=-\frac{x}{t^2}+\frac{1}{t}\left(\frac{x}{t}+\frac{1  }{t}\sin^2\frac{x}{t}\right)
    \frac{du}{dt}=\frac{1}{t^2}\sin^2\frac{x}{t}
    \frac{du}{dt}=\frac{\sin^2{u}}{t^2}
    Which you should be able to solve for u(t). Then just use x=ut to find x.

    --Kevin C.
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