Solving First Order Linear Inhomogenous Eq.

[Kalidor]

$$y'= \frac{y}{1+e^x}+e^{-x}$$

It's an easy first order linear inhomogenous eq. I solved it by hand with the formula that one can find anywhere AND with Mathematica, but when I take the derivative to check the solution it comes out wrong and it's freaking me out. Can anyone here post just the plain solution?

 

[Thaakisfox]

Just use the method of variating the constant, and you get:

<math>y(x) = C\frac{e^x}{1+e^x}-\frac{2+e^{-x}}{2(1+e^x)} </math>

 

[Thaakisfox]

<tex>y(x) = C\frac{e^x}{1+e^x}-\frac{2+e^{-x}}{2(1+e^x)} </tex>

 

[Thaakisfox]

oops used the wrong brackets for math mode, so here it is:

$$y(x) = C\frac{e^x}{1+e^x}-\frac{2+e^{-x}}{2(1+e^x)}$$

 

[blue_raver22]

Hello,

I understand your concern about the solution to this first order linear inhomogeneous equation. It is important to double check our solutions to ensure accuracy.

In this case, it is possible that there was an error in the differentiation process. I would recommend going back through the steps and checking for any mistakes.

Additionally, it may be helpful to use a different method of solving the equation, such as using an integrating factor or solving it as a separable differential equation. These methods may provide a different perspective and help confirm the solution.

If you are still having trouble, I would suggest seeking assistance from a colleague or mentor who may have more experience with this type of equation.

Best of luck with your solution.