Find the general solution of . Hint: Once known a solution , another solution is given by , with to determinate.
Attempt: I just don't know what method to use. I don't think variation of parameters work here since it's non linear.
Find the general solution of . Hint: Once known a solution , another solution is given by , with to determinate.
Attempt: I just don't know what method to use. I don't think variation of parameters work here since it's non linear.
Its a Cauchy-Euler equation.
So suppose that the equation has a solution of the form . Plugging this back into the ODE yields the characteristic equation .
In this particular problem, has the characteristic equation , which yields repeated roots.
So once you have , you suppose . It turns out that once you write the ODE in the form and substitute into it, we see that (this is the reduction of order technique as seen in my tutorial here.)
Then at this point, we see that is the second solution.
Can you continue with the problem?
Let , and . In this particular case, it has to be . Observe that and have different characteristic equations since they assume different solutions (the former assumes solutions of the form and the latter assumes solutions of the form )!
The former type of DE has the characteristic equation , where as the latter type has the equation .
I should have read better your first reply.
Ok, I get .
With , I reach that .
Thus the general solution would be of the form .
I hope it's right. And sorry, I didn't recognize it was a Cauchy-Euler equation.
No need to be sorry! Its just something that is useful to know down the road. Now, when we go ahead and find , we don't care much for the constants (they should drop out in the end anyways); so what where really after is that (or , depending on what notation is used where you're at). Thus, we see that the general solution is , which is what you have for your solution (but written in a slightly different way).
Hope that clarifies things!