1. ## [SOLVED] differential equation yrgent please

Use the integrating factor μ = ex to determine the general solution (in implicit form) of the differential equation

e^y +e^{−x} ln |x| + (e^y + y^2e^−x)dy/dx=0

How to solve this?

2. ## Integrating factor

Hello rajr
Originally Posted by rajr
Use the integrating factor μ = ex to determine the general solution (in implicit form) of the differential equation

e^y +e^{−x} ln |x| + (e^y + y^2e^−x)dy/dx=0

How to solve this?
Multiply both sides by $\displaystyle e^x$:

$\displaystyle e^xe^y + \text{ln}|x| + (e^xe^y + y^2)\frac{dy}{dx}=0$

Now if we differentiate $\displaystyle e^xe^y + \frac{1}{3}y^3$ with respect to $\displaystyle x$, we get:

$\displaystyle e^xe^y\frac{dy}{dx} + e^xe^y + y^2\frac{dy}{dx}$

$\displaystyle = e^xe^y + (e^xe^y + y^2)\frac{dy}{dx}$

So our equation is:

$\displaystyle \text{ln}|x| + \frac{d}{dx}(e^xe^y + \frac{1}{3}y^3) = 0$

$\displaystyle \Rightarrow e^xe^y + \frac{1}{3}y^3 = -\int \text{ln}|x|$

$\displaystyle \Rightarrow e^xe^y + \frac{1}{3}y^3 = -x \text{ln} |x| + x + c$