- #1

JPaquim

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## Homework Statement

[itex]\frac{d^2y}{dt^2} + t\frac{dy}{dt} + t^3y = e^t;\ \ \ y(0) = 0, \ \ y'(0) = 0[/itex]

Show that the solution is unique and has derivatives of all orders. Determine the fourth derivative of the solution at t = 0.

**2. The attempt at a solution**

I'm somewhat lost here... Trying to Laplace Transform it produces a third degree ODE, which doesn't really seem any simpler...

I guess I can calculate the fourth derivative at 0 by first calculating the second, by directly substituting the initial conditions, differentiating the quation and finding the third derivative, and differentiate it again to find the fourth... Doing it like so gives me [itex]y^{(4)} = 1[/itex]

However, I don't really know how to prove uniqueness nor C^∞ness... Any suggestions are welcome.

Cheers

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