I have the following differential equation:
$$ \frac{d^2y}{dt^2}+5\frac{dy}{dt}+6y=0 \\ y(0)=0\\ \frac{dy}{dt}(0) = 0\\ $$
To solve this differential equation I want to use the Laplace Transform. However, when solving this equation I get the following:
$$ s^2Y(s)+5sY(s)+6Y(s)=0\\ Y(s)(s^2+5s+6)=0\\ Y(s)=0 $$ Thus, according to Laplace there is no solution. However, using the standard way of solving second-order differential equations, there is a solution. Namely:
$$ y(t)=C_1e^{-3t}+C_2e^{-2t} $$
A hint of this solution can be found if we refactor the laplace expression: $$ s^2+5s+6\\ (s+3)(s+2) $$
Can you spot the mistake or misinterpretation in this derivation?
Thanks in advance!
Note that the solution does exist, its the constant function zero. If you use the fact that $y(0)=y'(0) =0$ to $$ y(t)=C_1e^{-3t}+C_2e^{-2t} $$ You'll get that $C_1=C_2=0$, so $y(t) =0$ and there is not any contradiction