Domain of definition of the solution of a Cauchy problem

Question

I need your help for the following Cauchy problem:

$$ y '(t) = (y (t) +9) ^ 2 \quad \text{with} \quad y (0) = 4 $$

I am trying to figure out on which domain, is the solution of this EDO defined?

I think the answer is entire $\mathbb {R} $. Is it correct ?

What I did:

Let's make the substitution $x(t) = (y(t) +9)$ Then the ODE is equivalent to: $x'(t)=x(t)^2$ I didn't go further because my goal is not to solve it but only to check the Domain of definition.

Thank you for your help!

Answer 1

Your guess is not correct, as you will see. We have techniques to know if the domain will be the whole $\mathbb{R}$ but here you may try to solve the equation to know where it is defined exaclty if it is not. Here is a way :

Doing the substition $z(t)=y(t)+9$ yeilds :

$z'(t)=z(t)^2$ with $z(0)=13$.

Then try to solve using separation of variables.

Answer 2

Try separation of variables:

$ \frac{dy}{(y+9)^2}=dx$.