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the question is:

$$(y^2+xy^3)\mathrm dx + (5y^2-xy+y^3\sin(y))\mathrm dy = 0$$

can any body tell me how to solve this linear equation?? when I tried to solve this the expression of integrating factor becomes too much difficult, may be i calculated it wrong... Any help will be appreciated. Thanks!

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Linear equation? The unknown function must be $x=x(y)$, then. – Giuseppe Negro Apr 20 '11 at 19:43
@Sadia: Looks to me to be an exact equation. – night owl Jul 4 '11 at 12:05

First I'd divide through by $y^2$ to make your life easier. Then the partial differential equation for the integrating factor becomes

$$xu+\frac{\partial u}{\partial y}(1+xy)=-\frac{1}{y}u+\frac{\partial u}{\partial x}(t-\frac{x}{y}+y\sin y)\;.$$

That happens to have a solution with $\frac{\partial u}{\partial x}=0$, so you can determine an integrating factor $u(y)$ from it.

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Partial differential equation? OP has a linear DE (in $x$), there's a formula for the integrating factor, no need to solve a PDE, is there? – Gerry Myerson Apr 26 '11 at 10:01
@Gerry: This is not what I'd call a linear DE (in $x$); it contains $x\mathrm dx$. If you mean the formula given e.g. here:…, I don't think it applies in this case; at least I don't see how to bring this equation into that form (after swapping $x$ and $y$). – joriki Apr 26 '11 at 10:19
sorry, must have posted while my brain was in neutral. – Gerry Myerson Apr 26 '11 at 12:59

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