# Can't a diffrential equation have same particular solutions analogous to polynomial roots

Our professor introduced the solution of a diffrential equations analogous to polynomials. (Finding roots vs finding functions which satisfied a particular set of operations). While solving 2nd order RLC circuits, I ran into a diffrential equation whose chracteristic roots were equal. My initial thought was that the diffrential equation had the same particular solution (just as a qudratic equation can have same roots). But after doing some googling I found out that it had a second solution (which was pretty unobvious)

Is it an established fact that an ODE have exactly $n$ independent diffrent solutions?

• I cant find a reference, but i remember that if you end up with multiple linearly dependent solutions $y_1 = y_2 = y_3 = e^{\lambda x}$ then you can create a new independent solution by multiplying the solutions with increasing powers of $x$ to get $y_1 = e^{\lambda x}, y_2 = x e^{\lambda x}, y_3 = x^2 e^{\lambda x}$ – DenDenDo Apr 10 '15 at 12:32
• Linear homogenous ODE of $n$-th order with constant coefficients has exactly $n$ linearly independent solutions. – TZakrevskiy Apr 10 '15 at 18:23