Toronto Math Forum
MAT2442013S => MAT244 MathTests => Quiz 3 => Topic started by: Victor Ivrii on February 27, 2013, 07:45:30 PM

4.3 p 239, # 4
Find a particular solution and then the general solution of the following ODE
\begin{equation*}
y'''y'= 2 \sin t .
\end{equation*}

General solution is the summation of the homogeneous and particular solutions. See attachment.

We start by solving $r^3r=0$ which gives that $r_1=0, r_2=1, r_3=1$.
Variation of parameters is not a good method to guess a particular solution here. You can try guessing that the particular solution is $Y_p=A\sin(t)+B\cos(t)$ or just look at the equation and deduce that $Y_p=\cos(t)$
So, general solution to the equation is
$Y_G=\cos(t)+c_1+c_2e^t+c_3e^{t}$.

Observing that the r.h.e. is an odd function and equation contains only odd derivatives we look for even solution: $y_p= A\cos(t)$ which makes easy problem even easier.