Expressing coefficients of a differential equation as decimal numbers

Question

I have a differential equation as such: $$\frac{d^2i}{dt^2}+7\frac{di}{dt}+12i(t)=36\delta(t-2) $$ Where $i(0)=0, i'(0)=0$. The solution is given by: $$i(t)=u(t-2)[Ae^{-3(t-2)}+Be^{-4(t-2)}] $$ Out of the solution, I need to express $A$ and $B$ as decimal numbers. Now, I can solve simpler differential equations, but the $\delta$ is completely throwing me off.

Answer 1

For $$i_0=Ae^{−3(t−2)}+Be^{−4(t−2)}$$ you need $i_0(2)=0$ and $i_0'(2)=36$ so that $$i(t)=u(t-2)i_0(t)$$ has the correct kink generating $36δ(t−2)$ in the second derivative. This gives the linear equations $$ A+B=0\\ -3A-4B=36 $$ which gives $A=-B=36$ as the solution.