# Toronto Math Forum

## MAT244-2018S => MAT244--Tests => Final Exam => Topic started by: Victor Ivrii on April 11, 2018, 08:44:51 PM

Title: FE-P4
Post by: Victor Ivrii on April 11, 2018, 08:44:51 PM
Find the general solution of the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = \hphantom{-}x +   y +\, \frac{e^{t}}{\cos(t)}\, ,\\
&y'_t = - x + y +\, \frac{e^{t}}{\sin(t)}\,.
\end{aligned}\right.
\end{equation*}
Title: Re: FE-P4
Post by: Tim Mengzhe Geng on April 12, 2018, 12:16:56 AM
First we start with the homogeneous system. The coefficient matrix is

A={
\left[\begin{array}{ccc}
1 & 1 \\
-1 & 1
\end{array}
\right ]},

The eigenvalues are

\lambda_1=1+i

\lambda_2=1-i

The eigenvector corresponding to $\lambda_1$ is

\xi^(1)=(1,i)^{T}

Then we can have the solution to the homogeneous system

(x,y)^T=c_1\cdot (e^t\cos(t), -e^t\sin(t))^T+c_2\cdot(e^t\sin(t), e^t\cos(t))^T

We can get a fundamental matrix from here

\Psi={
\left[\begin{array}{ccc}
e^t\cos(t) & -e^t\sin(t) \\
e^t\sin(t) & e^t\cos(t)
\end{array}
\right ]},

Note that we have

g(t)=(\frac{e^t}{\cos(t)},\frac{e^t}{\sin(t)})^T

And suppose

\Psi \mu^\prime=g(t)

We shoud have for the nonhomogeneous system

(x,y)^T=\Psi \mu

This is the method of Variation of Parameters. By plugging in and integration, we have

\mu_1(t)=c_3

\mu_2(t)=-\ln|cot(2t)+csc(2t)|+c_4

Finally we get the required solution according to (6), (9), (10) and (11)
Title: Re: FE-P4
Post by: Syed Hasnain on April 12, 2018, 01:37:56 PM
since the above solution is incomplete after μ(2)
I am attaching a copy of my solution...
Title: Re: FE-P4
Post by: Tim Mengzhe Geng on April 12, 2018, 03:32:57 PM
since the above solution is incomplete after μ(2)
I am attaching a copy of my solution...
I think your integration to $\mu_2(t)$may not be correct
Title: Re: FE-P4
Post by: Syed Hasnain on April 13, 2018, 02:37:46 AM
The integration of 1/(cos t sint) is ln(sint) - ln(cost) + c
and hence you get ln(tant) + c
Title: Re: FE-P4
Post by: Tim Mengzhe Geng on April 14, 2018, 02:42:24 AM
The integration of 1/(cos t sint) is ln(sint) - ln(cost) + c
and hence you get ln(tant) + c
$1/(costsint)=2/sin(2t)=2csc(2t)$
And by the integration of 2csc(2t),
we get
$-\ln|csc(2t)+cot(2t)|+c$
Title: Re: FE-P4
Post by: Tim Mengzhe Geng on April 14, 2018, 03:01:14 AM
The integration of 1/(cos t sint) is ln(sint) - ln(cost) + c
and hence you get ln(tant) + c
I'm so sorry that you're also correct  and the two result are the same since
$\tan\frac{x}{2}=\frac{\sin(x)}{1+cos(x)}$
and therefore
$-\ln|csc(2x)+cot(2x)|=\ln|tan(x)|$
Title: Re: FE-P4
Post by: Victor Ivrii on April 15, 2018, 03:26:19 AM
Solution is complete, but the answer must be written.

Also a simpler form $\ln(\tan(t))$ is preferable.

Finally \sin, \cos, and so on must be escaped by \ to provide upright letters and a proper spacing