MAT244-2013F > Quiz 3

Problem 2 (night sections)

**Victor Ivrii**:

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*}

**Ka Hou Cheok**:

\end{equation*}

The responding characteristic equation is $$r^3-r^2+r-1=0$$ 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.

**Yangming Cai**:

I am so impressed with your speed.

**Ka Hou Cheok**:

--- Quote from: Yangming Cai on November 06, 2013, 09:08:39 PM ---I am so impressed with your speed.

--- End quote ---

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.

**Victor Ivrii**:

I replaced

--- Code: --- sin , cos , sec

--- End code ---

by

--- Code: ---\sin, \cos, \sec

--- End code ---

and keyboard sign of integral by

--- Code: ---\int

--- End code ---

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