Compute output given input, transfer function and initial conditions The problem statement is

Consider a causal LTI system whose transfer function $H(s)$ is given as $$H(s)=\frac{s+2}{(s+3)(s+4)}$$ Compute the output $y(t)$ for an input $x(t)=e^{-2t}u(t)$ when $y(0)=1$ and $y’(0)=0$.

The solution is given as $y(t)=5e^{-3t} − 4e^{-4t}$.
Here is my attempt:

Laplace transform left side and right side and consider the initial condition


right side is 0 , and left side is below

so i think answer is below but the prof's answer is not match with my answer.
Your solution looks correct. In any case, the given solution $y(t)=5e^{-3t}-4e^{-4t}$, $t&gt;0$, must be wrong because it doesn't satisfy the initial condition on the derivative ($y'(0^+)=0$):
$$y'(t)=-15e^{-3t}+16e^{-4t},\qquad t&gt;0\tag{1}$$
So we have $y'(0^+)=1$. Your solution satisfies both initial conditions.