Calculating constant

Question

I am solving an initial value problem : $$\frac{dy}{dt}=y(-2t+\frac1{t})$$ After integrating I am stuck on: $$\log( y) = -t^2+\log( t) + c$$
The given initial condition is $y(0)=1$. Here the value of $t = 0$ and $\log(t)$ is undefined. How do I solve this?

Answer 1

There is no solution satisfying that initial condition. Note that the differential equation itself is undefined at $t=0$. It turns out that all solutions have $y(t) \to 0$ as $t \to 0$.

BTW, you made a small error in the integration. The right side should have been $-t^2 + \log(t) + c$. But that doesn't affect the conclusion.

Answer 2

First, you made a mistake, Integration yields $$ \ln y = -t^2+\ln t +c $$ and exponentiating yields $$ y = e^c t e^{-t^2} $$ and now you can apply the initial condition to see that you always get $1 = y(0) = 0$, and hence there is no solution corresponding to that initial condition.