Solving Differential equation with g(t) involving Dirac Delta function

Question

$y"+2y'-15y=6\delta(t-9), y(0)=-5, y'(0)=7$

Solutions to this differential equation is

$f(t)=\frac18e^{3t}-\frac18e^{-5t}, g(t)=\frac94e^{3t}+\frac{11}{4}e^{-5t},Y(s)=\frac{6e^{-9s}}{(s+5)(s-3)}-\frac{5s+3}{(s+5)(s-3)}$

So,$y(t)=6u_9(t)f(t-9)-g(t)$

Now, I want to verify initial condition y(0)=-5 S0,$y(0)=\frac68e^{-27}-\frac68e^{45}-\frac94-\frac{11}{4}=-2.62007032931E19-5$

But the initial condition is $y(0)=-5$, where i am wrong?

Answer 1

If $t<9$ then solution is $$y(t)=-\frac94e^{3t}-\frac{11}{4}e^{-5t}$$ This solution satisfies initial conditions $$y(0)=-5,\quad y'(0)=7$$