#### joshmccraney

Gold Member

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**1. The problem statement, all variables and given/known data**

Find the fundamental solution to the unbounded problem $$t u''(t) - u'(t) = \delta_0.$$

**2. Relevant equations**

Variation of parameters.

**3. The attempt at a solution**

I'm not sure how to use variation of parameters on this since it's an unbounded problem, so I'm not even trying to use it. The homogenous solution is ##u_h = c_1 t^2 + c_2##. So I'm thinking the fundamental solution should look something like

$$u =

\left\{

\begin{array}{ll}

c_1 t^2 + c_2 & t\leq 0 \\

c_3 t^2 + c_2 & t>0

\end{array}

\right.

$$

where I make both constant terms ##c_2## to ensure continuity at ##t=0##. But now I'm really not sure, is this even right?

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