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?
Hello rajrMultiply 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:
$\displaystyle \text{ln}|x| + \frac{d}{dx}(e^xe^y + \frac{1}{3}y^3) = 0$
$\displaystyle \Rightarrow e^xe^y + \frac{1}{3}y^3 = -\int \text{ln}|x|$
$\displaystyle \Rightarrow e^xe^y + \frac{1}{3}y^3 = -x \text{ln} |x| + x + c$
Grandad