Is there a Laplace transform for an unknown squared function? I was solving the differential equation
$$y'-y = e^x y^2$$
I know that this is a simple differential equation, and I would solve it with a simple change of the variable, however, I was wondering if I could use Laplace transform to make it easier and more direct.
I asked my professor, and she said she didn't know.
So is there a way to do $\mathscr{L}([y(x)]^2)$?
 A: $$y'-y = e^x y^2$$
Substitute $u=-\dfrac 1y$ then solve with Laplace transform:
$$u'+u=e^x$$
Otherwise you can't use Laplace transform with non linear DE.
A: Unless if you are able to make a clever substitution you cannot solve this ODE with the Laplace Transform.
The Laplace Transform is only used for solving linear equations, as it's a linear transform.
Trust me, i have tried solving nonlinear equations, i have failed.
What a Laplace Transform does to an ODE is that it basically finds the characteristic polynomial for a linear ODE.
$$ay''(t)+by'(t)+cy(t)=f(t)$$
$$\downarrow\mathscr{L}$$
$$a[s^2Y(s)-sy(0)-y'(0)]+b[sY(s)-y(0)]+cY(s)=F(s)$$
where $y(0), y'(0)$
are the intial conditions, as we use the Laplace Transform to see how a system dampens, so we always need those.
As you can see we get an equation of the form:
$$Y(s)=\xi(s)$$
where $\xi$ is a polynomial made of $s$-s.
For $y(0)=0, y'(0)=0$
The above equation will result in:
$$Y(s)=\dfrac{F(s)}{as^2+bs+c}$$
$$\downarrow\mathscr{L}^{-1}$$
Now after transforming back we'll get the solution to $y(t)$ and of course we get the same solution as if we use the classical method to solve the ODE.
Edit: For your case even with a substitution you'll still have to plug in initial conditions which sadly make this method kind of bad for general solutions (or you could assume y(0)=y'(0)=0, but that will give you a solution to an IVP)
A: Suppose $y$ is analytic at $0$, having power series
$$  y(x) = \sum_{k=0}^\infty a_k x^k  \text{.}  $$
Then
$$  y^2(x) = \sum_{n=0}^\infty \left( \sum_{i=0}^n a_i a_{n-i} \right) x^n  $$
and
$$  \mathcal{L}\{y^2\}(s) = \sum_{n=0}^\infty \left( \sum_{i=0}^n a_i a_{n-i} \right) \frac{n!}{s^{n+1}}  \text{,}  $$
wherever that series converges (if it converges anywhere).
I don't imagine this would ever be useful.  (Just how badly does applying the Frobenius method have to go to try taking a Laplace transform in the middle of it?)  But, a calculation can be done and there is a (formal) result.
One can certainly expand somewhere other that $0$, but this doesn't improve the situation.
