Toronto Math Forum
MAT2442013F => MAT244 MathTests => Quiz 3 => Topic started by: Victor Ivrii on November 06, 2013, 08:12:36 PM

Find the general solution of the given differential equation. Leave your answer in terms of one or more integrals.
\begin{equation*}
y'''y'' + y'y = \sec (t), \qquad \frac{\pi}{2} < t < \frac{\pi}{2}.
\end{equation*}

\end{equation*}
The responding characteristic equation is $$r^3r^2+r1=0$$ and we get $r_1=1, r_2=i, r_3=i$. So $$y_c=c_1e^t+c_2\cos(t)+c_3\sin(t)$$
$$W=e^t((\sin^2(t)+\cos^2(t)\sin(t)\cos(t))(\sin^2(t)\cos^2(t)\sin(t)\cos(t)))=2e^t$$
$$W_1=\cos^2(t)+\sin^2(t)=1\\
W_2=e^t(\sin(t)\cos(t))\\
W_3=e^t(\sin(t)\cos(t))$$
$$u_1=\int \frac{(\sec(t))(1)}{2e^t}dt\\
u_2=\int \frac{(\sec(t))(e^t(\sin(t)\cos(t))}{2e^t}dt\\
u_3=\int \frac{(\sec(t))(e^t(\sin(t)\cos(t))}{2e^t}dt$$
$$y=y_c+y_1u_1+y_2u_2+y_3u_3
=c_1e^t+c_2\cos(t)+c_3\sin(t)+\\
e^t\int \frac{(\sec(t))(1)}{2e^t}dt+
\cos(t)\int \frac{(\sec(t))(e^t(\sin(t)\cos(t))}{2e^t}dt+
\sin(t)\int \frac{(\sec(t))(e^t(\sin(t)\cos(t))}{2e^t}dt$$
As the question stated my answer can be in terms of one or more integrals, hopefully I can stop here.

I am so impressed with your speed.

I am so impressed with your speed.
I would take it as a compliment. Thanks.
I'm impressed and appreciate your results of the integrals. I was too lazy to integrate them.

I replaced
sin , cos , sec
by \sin, \cos, \sec
and keyboard sign of integral by \int

I replaced sin , cos , sec
by \sin, \cos, \sec
and keyboard sign of integral by \int
Thanks, Prof Victor. I'll use them well next time.