Very small question, help appreciated

[I_love_everyone]

Hi everyone,
I hope you're alright if you're reading this.
Simple question about differential equation :

Concerning the solutions of the heterogeneous equation, do we all agree on the fact that different methods will lead to different solutions ?
I humbly thank you if you take the time to give an appropriate response.

(I'm not an english native, so i'm not very familiar with math vocabulary in english. I hope that I have been clear enough).

 

[nasi112]

Do you mean a non-homogeneous equation? Something like this:

\(\displaystyle \frac{dy}{dx} + y = \sin x\)

Whatever different method you will use, eventually you will arrive to the same general solution if you combine the complementary solution and the particular solution.

Finally, to make the general solution unique, you will need to have side conditions or sometimes called initial conditions.

 

[I_love_everyone]

Yes ! It’s exactly what I meant and by chance, your example was the exact same equation I was trying to solve.
I was using different methods to get the particular solution and didn’t realize that I made many mistakes along the way, leading to different answers. But I was still having a little doubt about it, and you cleared that for me :
Thank you !

I hope you have a great day.

 

[nasi112]

You're welcome