Originally Posted by

**HallsofIvy** Are there **any** examples of differential equations problems where "Laplace transform method" is worth doing?

Here, the first thing I would do is see that y''+ 6y'+ 9y= 0 has characteristic equation $\displaystyle r^2+ 6r+ 9= (r+ 3)^2= 0$ and so has r= -3 as a double root. That tells us that the general solution can be written as $\displaystyle y(t)= C_1e^{-3t}+ C_2te^{-3t}$. From that, $\displaystyle y'(t)= -3C_1e^{-3t}+ C_2e^{-3t}- 3C_2te^{-3t}$.

All that remains is to find $\displaystyle C_1$ and $\displaystyle C_2$ so that $\displaystyle y(0)= C_1= -1$ and $\displaystyle y'(0)= -3C_1+ C_2= 7$.

I have always disliked the Laplace transform method!