# Differential equation problem

• January 30th 2013, 10:21 AM
patzer
Differential equation problem
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
• January 30th 2013, 12:17 PM
FernandoRevilla
Re: Differential equation problem
Quote:

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.
• January 30th 2013, 03:48 PM
patzer
Re: Differential equation problem
Hello sir,

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?