# Thread: Seperation of Variables (inc cos and exp)

1. ## Seperation of Variables (inc cos and exp)

Hi all,
Problem below:

Solve the following differential equation

$\displaystyle \frac{dy}{dx} = \frac{cos(2x+1) + e^{3x}}{y}$

I can re arrange this into:

$\displaystyle \frac{dy}{y} = cos(2x+1) + e^{3x}dx$

From there integration brings:

$\displaystyle \ln|y| + c = \frac{sin(2x+1)}{2} + \frac{e^{3x}}{3} + c$

then multiplying by $\displaystyle e^{x}$:

$\displaystyle y + c = e^{\frac{sin(2x+1)}{2} + \frac{e^{3x}}{3} + c}$

Now I'm stuck. I can't see any way to simplify the $\displaystyle e^{()}$ value on the right. It doesn't really feel solved to me... Is it?

2. Hello, isp_of_doom!

Your first step is wrong . . .

$\displaystyle \frac{dy}{dx} \:=\: \frac{\cos(2x+1) + e^{3x}}{y}$

We have: .$\displaystyle y\,dy \;=\;\left[\cos(2x+1) + e^{3x}\right]\,dx$

Integrate: .$\displaystyle \tfrac{1}{2}y^2 \;=\;\tfrac{1}{2}\sin(2x+1) + \tfrac{1}{3}e^{3x} + C$

. . . . . . . . .$\displaystyle y^2 \;=\;\sin(2x+1) + \tfrac{2}{3}e^{3x} + C$