Differential equation problem

**patzer** · January 30th 2013, 10:21 AM

Hello,

I'm trying to find the integration factor $\lambda(x,y)$ of this differential equation: $(siny-3x^2cosy)cosydx+xdy=0$ , with exception that $\lambda$ being a function, dependent in only one variable, x or y.

Any ideas on solving this equation?

Thanks

**FernandoRevilla** · January 30th 2013, 12:17 PM

Originally Posted by patzer

Hello,

I'm trying to find the integration factor $\lambda(x,y)$ of this differential equation: $(siny-3x^2cosy)cosydx+xdy=0$ , with exception that $\lambda$ being a function, dependent in only one variable, x or y.

Any ideas on solving this equation?

Thanks

Hint: If $Pdx+Qdy=0$ and $\lambda(x,y)=\mu (z)$ , the equality $(\mu P)_y=(\mu Q)_x$ sometimes (only sometimes) allows to predict the form of $z$ . Take into account that in general, if we don't know a priori the form of the integration factor, there is no a general method to find it.

**patzer** · January 30th 2013, 03:48 PM

Hello sir,

Thank you for your help.

This is my progress so far:

I know that $\frac{d\lambda}{\lambda}=-\frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{P}dy$ where $\frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{P}$ = $\frac{3x^2sin(2y)+cos(2y)-1}{3x^2(cosy)^2-cosysiny}$

My problem now is with this expression $\frac{3x^2sin(2y)+cos(2y)-1}{3x^2(cosy)^2-cosysiny}$ , I simply don't know how to simplify it.
Wolfram Alpha gives $2tan(y)$ as the answer. Can you help me on this point?

Thank you again.

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