Norwin Dole's question at Yahoo! Answers about a differential equation

[Fernando Revilla]

Here is the question:

How to obtain the general solution of ye^(2x) dx = (4+e^(2x))dy?
Separation of Variables - First Order Differential Equation
Please show solutions. Thanks

Here is a link to the question:

How to obtain the general solution of ye^(2x) dx = (4+e^(2x))dy? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

We can solve it by separation of variables:

[tex]\dfrac{dy}{y}=\dfrac{e^{2x}\;dx}{4+e^{2x}}[/tex]

Integrating both sides:

[tex]\ln |y|=\dfrac{1}{2}\ln|4+e^{2x}|+C=\ln \sqrt{4+e^{2x}}+C[/tex]

Equivalently,

[tex]y=C_1\left(\sqrt{4+e^{2x}}\right)[/tex].