A second order nonlinear ODE? Whoa!

Homework Statement

I'm not sure if this is actually solvable, or a typo on my homework... but here's the problem in question:

Solve the ODE:
$$\frac{d^{2}y}{dt^{2}} + t^{2} \frac{dy}{dt} + y^{2} = 0, y(0)=0, y'(0)=0$$

Attempt at solution
I've been stumped on where to even start with this one, but I attempted to substitute $$v=y'$$. That didn't get me very far though, so any tips on where to start would be appreciated!

Typo-wise, I'm also wondering if perhaps the $$y^{2}$$ in the equation is actually just supposed to be a y... which would make it significantly easier.

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gabbagabbahey
Homework Helper
Gold Member
This looks almost like a trick question....given your initial conditions, what will $y''(0)$ be? What then would you expect $y'(0+dt)$ and $y(0+dt)$ to be?...Do you see where this is going?

HallsofIvy
Homework Helper

Homework Statement

I'm not sure if this is actually solvable, or a typo on my homework... but here's the problem in question:

Solve the ODE:
$$\frac{d^{2}y}{dt^{2}} + t^{2} \frac{dy}{dt} + y^{2} = 0, y(0)=0, y'(0)=0$$

Attempt at solution
I've been stumped on where to even start with this one, but I attempted to substitute $$v=y'$$. That didn't get me very far though, so any tips on where to start would be appreciated!

Typo-wise, I'm also wondering if perhaps the $$y^{2}$$ in the equation is actually just supposed to be a y... which would make it significantly easier.
Actually, this whole problem is trivial! What is the easiest possible function that satisfies y(0)= 0, y'(0)= 0? Can you show that that is a solution to the differential equation? And, of course, by the "existence and uniqueness" theorem, it is the solution.

Ooooooh....

Okay, that makes perfect sense! I was making it out to be a lot harder than it was, and wasn't trying to solve it by inspection, but after looking at it with your hints, it's very clear.

Thank you both!

Just out of curiosity, if the initial conditions weren't given, would it be solvable analytically?