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How do i solve equations like, $f'(x) = f^3 + f^2 + f$ using laplace transforms? Any help would be appreciated.....

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Do you know what laplace transformation is? If you do then apply this transformation to both sides of equation. – nikita2 Sep 13 '12 at 16:21
If you have a transform T with sufficiently nice properties (e.g., doing things like relating $T(f')$ to $T(f)$ or $T(fg)$ to $T(f)$ and $T(g)$), then a differential equation for $f$ often gives rise to a differential or algebraic equation for $T(f)$. If one can solve this new equation and apply the inverse transform to the result, it yields a solution of the original equation. Try looking at the properties of the Laplace transform to see if you can perform the first step mentioned on your example. – Aaron Sep 13 '12 at 16:24
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Generally the strength of the Laplace transform is in dealing with linear systems, since convolutions become multiplications. In this case, multiplications will become convolutions of sorts, so I doubt that the Laplace transform will simplify your life with this problem... – copper.hat Sep 13 '12 at 16:27
@Ali are you the RHS is not $f^{(3)} + f^{(2)} + f$? (ie. $f''' + f'' + f$) – user2468 Sep 14 '12 at 1:28
I think your best bet is the familiar separation technique: $\int df/(f^3+f^2+f) = \int dt$. The integral on the left is not the nicest imaginable, but with partial fractions you can do it. The result will be in implicit form, $\Phi(f)=t+C$. – user31373 Sep 14 '12 at 3:19
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