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

AI Thread Summary
The discussion centers on solving the differential equation ye^(2x) dx = (4+e^(2x))dy using the method of separation of variables. The solution involves rearranging the equation to isolate dy/y and integrating both sides. The integration results in ln|y| = (1/2)ln|4+e^(2x)| + C, leading to the general solution y = C_1(√(4+e^(2x})). This approach effectively demonstrates how to derive the general solution for the given first-order differential equation. The thread provides a clear example of applying separation of variables in differential equations.
Fernando Revilla
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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 :D

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.
 
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We can solve it by separation of variables:

\dfrac{dy}{y}=\dfrac{e^{2x}\;dx}{4+e^{2x}}

Integrating both sides:

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

Equivalently,

y=C_1\left(\sqrt{4+e^{2x}}\right).
 
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