Find the general solution of
Hi
First, solve the homogeneous equation $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2} +4 \frac{\mathrm{d}y}{\mathrm{d}x} +3y=0$. It implies : writing the characteristic equation and finding its root. Then, using these roots, you can deduce an expression of the solution $\displaystyle y_h$.
Then, find a particular solution $\displaystyle y_p$ of $\displaystyle \frac{\mathrm{d}^2y}{\mathrm{d}x^2} +4 \frac{\mathrm{d}y}{\mathrm{d}x} +3y=2\exp(-3x)$. It can be achieved by showing that $\displaystyle y_p(x)=\lambda \exp(-3x)$ is a solution for some value of $\displaystyle \lambda$. (usually, if the second member is a polynomial of $\displaystyle \exp x$ searching for a particular solution of this form works well)
The solution of the complete equation will be the sum $\displaystyle y_h+y_p$.