# Solving a Cauchy problem

Given the equation $$y' = e^t \sqrt[3]{y^2}$$

(a) consider the related Cauchy problem with $y(t_0)=y_0$. What $P(t_0,y_0)$ ensures the problem has a unic solution?

(b) find the general integral of the given equation.

I'm not able to answer the first question. I know that if $f(t,y(t))=e^t \sqrt[3]{y^2}$ is lipschizt with respect to $t$ and continuos with respect to $t$, then there exists an interval $I$ centered in $t_0$ where the solution is unic. But how can I apply this theorem? Also, do I just need this theorem?

Oss: I shall suppose $y(t)$ of class $C^1$ for regularity of $f$.

Plus, how can I solve the differential equation? Do I need some other hypothesis on $y(t)$?

Thanks

• for b) write the equation as $$\frac{dy}{(y^2)^{1/3}}=e^tdt$$ – Dr. Sonnhard Graubner Jan 21 '17 at 19:18