I want to solve the differential equation $$\dddot{y}-4\ddot{y}+5y-2=\sin t$$ but since now I only know how to solve first order differential equations using variation of parameters or separating variables.
What is the general procedure to solve such an equation?
In fact you must separate the answer by two: homogeneous and private and then add them up to obtain general answer since the ODE is linear. The homogeneous answer is the answer of the following ODE $$y'''-4y''+5y=0$$the characteristic equation is$$r^3-4r^2+5=0$$with roots $r_1,r_2,r_3$ and the homogeneous answer would be $$y_h=Ae^{r_1t}+Be^{r_2t}+Ce^{r_3t}$$for arbitrary constants $A$, $B$ and $C$. The private answer is that of the following ODE:$$y'''-4y''+5y=2+\sin t$$which according to sinusoid form of input should be of form:$$y_p=P\sin t+Q\cos t+R$$by substitution in the ODE we get:$$(9P+Q)\sin t+(9Q-P)\cos t+5R=2+\sin t$$which yields to $P=\dfrac{9}{82},Q=\dfrac{1}{82}$ and $R=\dfrac{2}{5}$. Therefore the private answer is $$y_p=\dfrac{9}{82}\sin t+\dfrac{1}{82}\cos t+\dfrac{2}{5}$$and the general answer would be obtained adding up the homogeneous and private answers as following:$$y_g=y_h+y_p=Ae^{r_1t}+Be^{r_2t}+Ce^{r_3t}+\dfrac{9}{82}\sin t+\dfrac{1}{82}\cos t+\dfrac{2}{5}$$