Hello.

Problem: Find the solution of the differenial equation that satisfies the given intial condition.

$\displaystyle \left( 2y + e^{3y} \right) \frac{dy}{dx} = x cos(x)$

The intial condition:$\displaystyle y(0)=0$.

My Solution:

I rewrite it as:

$\displaystyle \left( 2y + e^{3y} \right)dy=xcos(x)dx$

By integrating both sides and using integration by parts for the right side, I obtained:

$\displaystyle y^2+\frac{1}{3}e^{3y}=xsin(x)+cos(x)+C$.

The problem here is that I can not solve the last equation for $\displaystyle y$ so that I can use the intial condition to find the particular solution.

Any help?