Variation of parameters for higher-order linear ODE

Question

I am rather unfamiliar with proper protocol and etiquette for asking for help solving homework problems, so I apologize if I do anything wrong. I've spent an exorbitant amount of time today trying to wrap my head around variation of parameters for higher-order linear ODEs, and I am at my wit's end. Any help would be greatly appreciated. In particular I am defeated by the fourth-order linear ODE $$y^{(4)}+2y^{''}+y=sin(t).$$ According to my textbook, the solution is $$y=c_1cos(t)+c_2sin(t)+c_3tcos(t)+c_4tsin(t)-\frac{1}{8}t^2sin(t).$$ I have been working with the formula $$Y(t)=\sum_{m=1}^ny_m(t)\int_{t_0}^{t}\frac{g(s)W_m(s)}{W(s)}ds$$ to find a particular solution $Y(t)$ to combine with $y_c(t)$ for a general solution $y(t)$. I am fine arriving at $y_c(t)$, but working towards $Y(t)$ I end up with the Wronskian $$W=\begin{vmatrix}cos(t)&sin(t)&tcos(t)&tsin(t)\\-sin(t)&cos(t)&cos(t)-tsin(t)&sin(t)+tcos(t)\\-cos(t)&-sin(t)&-2sin(t)-tcos(t)&2cos(t)-tsin(t)\\sin(t)&-cos(t)&-3cos(t)+tsin(t)&-3sin(t)-tcos(t)\end{vmatrix},$$ which stumps me. I tried instead using complex numbers to avoid dealing with so much trig, and had $e^{it},e^{-it},te^{it},\text{ and }te^{-it}$ filling out the first row in $W$, but that also got me nowhere. MatLab tells me the Cramer's rule determinants are $$\begin{align}W_1&=4cos^4(t)-4sin^4(t)-8cos^2(t)sin^2(t)\\W_2&=2cos(t)(cos^t(t)+sin^2(t))\\W_3&=-2tsin^3(t)-2cos^3(t)-2cos(t)sin^3(t)-2tcos^2(t)sin(t),\text{ and }\\W_4&=2sin^3(t)+2cos^2(t)sin(t)\end{align}$$I suspect I have gone astray somewhere in this mess, because I have no idea how I would get $\frac{-1}{8}t^2sin(t)$ out of all that. I am just so inordinately stressed right now, I could really use some assistance. Also, variation of parameters is the required method for me to use. I haven't tried undetermined coefficients because that method was last week's assignments. Less urgent because it is an uncollected exercise, but I am similarly unable to solve $y^{'''}-y^{''}+y^{'}-y=e^{-t}sin(t).$

Answer 1

Note that your Wranskian $$W=\begin{vmatrix}cos(t)&sin(t)&tcos(t)&tsin(t)\\-sin(t)&cos(t)&cos(t)-tsin(t)&sin(t)+tcos(t)\\-cos(t)&-sin(t)&-2sin(t)-tcos(t)&2cos(t)-tsin(t)\\sin(t)&-cos(t)&-3cos(t)+tsin(t)&-3sin(t)-tcos(t)\end{vmatrix}$$simplifies if you add the first row to the third and add the second row to the fourth row.

You will get $$W=\begin{vmatrix}cos(t)&sin(t)&tcos(t)&tsin(t)\\-sin(t)&cos(t)&cos(t)-tsin(t)&sin(t)+tcos(t)\\0&0&-2sin(t)&2cos(t)\\0&0&-2cos(t)&-2sin(t)\end{vmatrix}=4$$