Superposition Principle and Fundamental Set Difference?

[Integrate]

So as I understand, the super position principle means that if we have several particular solutions to a DE we can add them together to get a general solution.

Am I missing anything there? Like [math]y_1\:=\:x^2\:and\:y_2\:=\:e^x\:\:->\:y_g=c_1x^2+c_2e^x[/math]

I don't know if there are any limitations here, but it seems that any solution can be added together to create a general solution respective to a DE.

Now for the fundamental set this is taken a little bit further. If those solutions are linearly independent, then they are said to be a fundamental set.

I guess my question is if linear independence has any other effect on the super position principle.


Like, do I need to check linear independence for a set of solution to know if the superposition principle to work? or will it work with any solutions regardless of independence?


Let me know if my understanding of these concepts is correct.

 

[mario99]

Before we answer your question, we want you to give us a scenario for a differential equation in which there is at least one dependent solution.

 

[Integrate]

[math]y_1=e^x, y_2=2e^x \\ y'_1=e^x , y'_2=2e^x \\ y"_1=e^x , y"_2=2e^x \\ y"_1-y'_1+y_1=e^x \\ y"_2-y'_2+y_2= 2e^x \\ y"-y'+y = c_1e^x+c_22e^x[/math][math]\\ c_1 = -2 \\ c_2 = 1[/math]

 

[mario99]

If your differential equation is:

[imath]\displaystyle y'' - y' + y = e^x - 2e^x[/imath]

It means that your differential equation is:

[imath]\displaystyle y'' - y' + y = -e^x[/imath]

If your particular solutions are:

[imath]y_1 = c_1e^x[/imath] and [imath]y_2 = c_2e^x[/imath]

It means that your particular solution is:

[imath]y_p = (c_1 + c_2)e^x[/imath]

Which is not a scenario where the particular solution is dependent!