# Uniqueness of a solution of differential equation

I have been trying to deal with this differential equation

$y'=\frac{2}{t}y+t^2e^t, y(0)=1, 1\leq t\leq2$, and the problem asks to show that the solution is unique. I think this shouldn't be correct.

Firstly, existence and uniqueness theorem doesn't apply here since the initial point $0$ doesn't lie in the interval $[1,2]$.

Secondly, If I solve this differential equation without Initial value the solution is $y(t)=c_1t^2+t^2e^t$, which has no solution if i take $y(0)=1$.

Any thoughts about the problem?

• Can you ask back if the initial condition was not meant to be $y(1)=1$? It really makes no sense to put it in the singular point outside the interval of definition. – LutzL May 5 '15 at 9:40
• That's what I have been thinking. If the initial condition is what you've suggested then there is no problem at all. – Mathematician May 5 '15 at 13:48
• If you can't ask back, then write it down exactly like that. First the general solution and clearly separate the discussion of initial values, the impossibility of $t=0$ (can be seen from the original equation, without solving anything) and the proposed correction of the task. – LutzL May 5 '15 at 15:58