The roots of a cubic auxiliary equation are $_1 = 4$ and $_2 = _3 = 5$. What is the corresponding homogeneous linear differential equation?

Question

Is your solution unique?

My try:

$m-4=0, m-5=0 \text { and } m-5=0$

$(m-4)(m-5)^{2}=0$

$(m-4)(m^{2}-10m+25)=0$

$m^{3}-10m^{2}+25m-4m^{2}+40m-100=0$

$m^{3}-14m^{2}+65m-100=0$

$d^{3}y/dx^{3}-14d^{2}y/dx^{2}+65dy/dx-100y=0$

Is this correct ?

How to check uniqueness ?

Thanks.

Answer 1

If you have an ODE of the form: $$\sum^n_{i=1}a_iy^{(i)}=0$$ and make the substitution $y=\mu\exp(\lambda x)$ you will end up with the following: $$\sum_{i=1}^na_i\lambda^i=0$$ rearranging this you will get a polynomial, the roots of which you already know. Working back from this you can get the the order of the derivatives and the coefficients $a_i$