Toronto Math Forum
MAT2442013S => MAT244 MathTests => Quiz 3 => Topic started by: Victor Ivrii on February 27, 2013, 07:43:47 PM

4.4 p 244, # 1
Find a particular solution by method of variation of parameters and then the general solution of the following ODE :
\begin{equation*}
y'''+y'=\tan t,\qquad \frac{\pi}{2}<t<\frac{\pi}{2}.
\end{equation*}

Here is my solution:
where is says c1 in the final answer, replace that with c0 haha

We first solve the characteristic polynomial: $r^3+r=0$ which yields $r_1=0, r_{2,3}=+ i$. So, solution to the general homogeneous equation is
$$y_g=u_1+u_2\cos(t)+u_3 \sin(t)$$
Idea in method of parameters is to let $u_1,u_2,u_3$ to be functions and then plot the $y_g$ into the equation (in this case compute it's first and third derivative), solving for $u_1,u_2,u_3$.
This can be done, but textbook page 242 gives you a nice final formula: $$u'_m=\frac{gW_m}{W}$$
So, we use this  maybe this was the reason why I finished 7 minutes earlier.
It turns out that $W(1,\cos(t), \sin(t))=1$. $W_m$ is just the same wronskian but the mth column is replaced by the column $(0,0,1)$. With this definition
$$W_1=1, W_2=\cos(t), W_3=\sin(t)$$
Using the previously stated formula and the fact that in this case $g=\tan(t)$:
$$u_1'=\tan(t) \implies u_1=\ln(\cos(t))$$
$$u_2'=\tan(t)\cos(t)=\sin(t)\implies u_2=\cos(t)$$
$$u_3'=\tan(t)\sin(t)=\left(\frac{1\cos^2(t)}{\cos(t)} \right)=\left(\frac{1}{\cos(t)}\cos(t)\right)\implies
u_3=\ln(\sec(t)+\tan(t))+\sin(t)$$
The third integral was the hardest. But in the homework problems you also had to compute $\int \frac{dt}{\cos(t)}$ which is surprisingly difficult. So, I had worked on the integral before and remembered the solution.
Now, we just have to plug $u_1,u_2,u_3$ back to the general solution of homogeneous equation $y_g$ and we obtain that the particular solution is:
$$Y_p=\ln(\cos(t))+\cos^2(t)+\sin^2(t)\sin(t)\ln(\sec(t)+tan(t))=\ln(\cos(t))+1\sin(t)\ln(\sec(t)+\tan(t))$$
General solution is therefore
$$Y_G=Y_p+c_1+c_2\cos(t)+c_3 \sin(t)$$