Solution of $ y''+5y'+6y=0$ with initial condition , $y(0)=2, y'(0)=3$ using laplace transform The below equation $(1)$ is second-ordinary differential equation, I have got the solution of it using standard method which is :$y(x)=e^{-3x}(-7+9e^x)$ , now my question is to solve  equation  $(1)$ using laplace transform :
$ y''+5y'+6y=0, y(0)=2, y'(0)=3\tag{1}$
 A: Let $Y(s)$ be the laplace transform of $y(x)$. The first step is to take the Laplace transform of both sides. Recall that the Laplace transform is linear, and acts on derivatives in the following way:
\begin{align*}
\mathcal{L}(y')(s) &= sY(s) - y(0) \\
&= sY(s) - 2 \\
\mathcal{L}(y')(s) &= s^2Y(s) - sy(0) - y'(0) \\
&= s^2Y(s) - 2s - 3
\end{align*}
Therefore, taking the Laplace transform of both sides yields,
$$s^2Y(s) - 2s - 3 + 5(sY(s) - 2) + 6Y(s) = 0$$
Expanding and collecting terms of $Y(s)$ yields
$$(s^2 + 5s + 6)Y(s) = 2s + 13 \implies Y(s) = \frac{2s + 13}{(s + 2)(s + 3)}.$$
Applying a partial fractions decomposition further gives us
$$Y(s) = \frac{9}{s + 2} + \frac{-7}{s + 3}.$$
We have solved for the Laplace of $y(x)$, so we need to apply the inverse Laplace transform. Consulting a standard transform table, this gives us an answer of
$$y(x) = 9e^{-2x} - 7e^{-3x}.$$
A: $$y''+5y'+6y=0, y(0)=2, y'(0)=3 \implies$$
$$[s^2F(s)-sy(0)-y'(0)]+5[sF(s)-y(0)]+6F(s)=0 \implies $$
$$(s^2+5s+6)F(s)=2s+13 \implies $$
$$F(s)=\frac{2s+13}{s^2+5s+6}=\frac{9}{s+2}-\frac {7}{s+3}\implies$$
$$ y(t) = 9e^{-2t}-7e^{-3t}$$
A: this way dosent make things simple but it's still funny..
$$y''+5y'+6y=0, y(0)=2, y'(0)=3$$
$$y''+2y'+3y'+6y=0$$
Substitute $w=y'+2y$
$$w'+3w=0$$
$$\mathcal{ L}(w'+3w)=sF(s)-w(0)+3F(s)=0$$
Since $w(0)=y'(0)+2y(0)=7$
$$sF(s)-7+3F(s)=0$$
$$F(s)=\frac 7 {s+3}$$
$$w=7e^{-3x}$$
$$y'+2y=7e^{-3x}$$
$$\mathcal{L}(y'+2y)=7\mathcal{L}(e^{-3x})$$
$$sF(s)-y(0)+2F(s)=\frac 7 {s+3}$$
$$(s+2)F(s)=\frac 7 {s+3}+2$$
$$F(s)=\frac 7 {(s+2)(s+3)}+\frac 2 {s+2}=\frac 9 {s+2}-\frac 7 {s+3}$$
$$y(x)=9\mathcal{L^{-1}}(\frac 1 {s+2})-7\mathcal{L^{-1}}(\frac 1 {s+3})$$
$$\boxed{y(x)=9e^ {-2x}-7e^{-3x}}$$
