Would this be considered sufficient/correct proof for a second solution of 2nd order homogeneous linear differential equations? I skipped some steps and explanations here just for brevity:
$$(1) \ \ \ ay''+by'+cy=0$$ The characteristic equation of $(1)$ is: $$(2) \ \ \ ar^{2}+br+c=0$$ Assuming $b^{2}-4ac=0$ then $(2)$ has roots $r_1=r_2=r=\frac{-b}{2a}$, and so $y_1=e^{rt}$ is a solution,
$$\implies\ a\frac{d^{2}}{dt^{2}}\left[ e^{rt} \right]+b\frac{d}{dt}\left[ e^{rt} \right]+ce^{rt}=0$$ Multiplying both sides by $\frac{d}{dr}$ we get: $$a\frac{d^{2}}{dt^{2}}\left[ te^{rt} \right]+b\frac{d}{dt}\left[ te^{rt} \right]+cte^{rt}=0$$ $\implies \ y_{2}=te^{rt}$ is a second solution satisfying $(1)$.