### Author Topic: TT1--Problem 3  (Read 8589 times)

#### Victor Ivrii

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##### TT1--Problem 3
« on: February 13, 2013, 10:39:58 PM »
Find the general solution for equation
\begin{equation*}
y'' + 4y'+5y =t e^{-2t}+ e^{-2t}\cos(t).
\end{equation*}

#### Devin Jeanpierre

• Jr. Member
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##### Re: TT1--Problem 3
« Reply #1 on: February 13, 2013, 11:30:33 PM »
Funny story: after I posted this I also checked my work against wolfram alpha, which said I got it wrong for the homogeneous part. I spent some time trying to spot any error, but there isn't any. I am smarter than the machine!

(You can press "show step by step solution" and wolfram alpha will come up with what I came up with, so that's why I'm so confident. I filed a bug report, maybe someday this will be fixed...)

$\renewcommand{Re}{\operatorname{Re}}$

Here we use the method of undetermined coefficients. First we find the characteristic equation and its derivative (which will definitely be useful):

$$Q(r) = r^2 + 4r + 5$$
$$Q^\prime(r) = 2r + 4$$

Now we solve for $te^{-2t}$, for which our exponent is $r_1 = -2$.

$$Q(r_1) = 4 - 8 + 5 = 1$$
$$Q^\prime(r_1) = -4 + 4 = 0$$

So we want to use $t e^{-2t}$. If we plug that in we get $L[t e^{-2t}] = e^{-2t}(t Q(-2) + Q^\prime(-2)) = t e^{-2t}$. This is what we want, so we're done the first part.

Now we solve for $e^{-2t} cos(t)$, for which the exponent is $r_2 = -2 + i$.

$$Q(r_2) = (-2+i)^2 + 4(-2 + i) +5 = 4 - 4i - 1 - 8 + 4i + 5 = 0$$
$$Q^\prime(r_2) = 2(-2+i) + 4 = -4 + 2i + 4 = 2i$$

So we want to use $t e^{(-2 + i)t}$. If we plug that in, we get $L[t e^{(-2 + i)t}] = e^{(-2 + i)t}(t Q(-2 + i) + Q^\prime(-2 + i)) = (2i)e^{(-2 + i)t}$.

We don't want that 2i, so let's divide it out in the input. It turns out that $1/2i = i/-2 = -i/2$. So $L[\frac{-i}{2} t e^{(-2 + i)t}] = e^{(-2 + i)t}$, which is nearly right.

Note:
$$\Re e^{(-2 + i)t}) = \Re (e^{-2t}e^{it}) = \Re (e^{-2t}(\cos t + i \sin t)) = e^{-2t} \cos t$$.

Since $L$ is linear, if we take only the real part of the input, we'll get only the real part of the output. So we need to compute the real part of $\frac{-i}{2} t e^{(-2 + i)t}$.

To do that let's multiply it out:

$$\frac{-i}{2} t e^{(-2 + i)t}$$
$$\frac{-i}{2} t e^{-2t}(\cos t + i \sin t)$$
$$t e^{-2t}(\frac{-i}{2}\cos t + \frac{-i}{2}i \sin t)$$
$$t e^{-2t}(\frac{-i}{2}\cos t + \frac{1}{2}\sin t)$$

The real part of that is $\frac{1}{2}t e^{-2t}\sin t$, so that $L[\frac{1}{2}t e^{-2t}\sin t] = e^{-2t} \cos t$.

So then we put our two subproblem solutions together and we get a particular solution, thanks to the linearity of $L$. Our particular solution is:

$$Y(t) = t e^{-2t} + \frac{1}{2}t e^{-2t}\sin t$$

What remains is to compute a general solution for the homogenous part, and we can combine that to find the general solution for the non-homogenous ODE.

During the above work we found that $-2 + i$ was a root of the characteristic equation. So the conjugate is also a root, $-2 - i$, and we can take the real and imaginary parts of $e^{-2 \pm i}$ to find the solution, which is:

$$y_{gen[homogeneous]}(t) = c_1 e^{-2t}\cos(t) + c_2 e^{-2t}\sin (t)$$

So the general solution for the nonhomogeneous ODE is:

$$y(t) = c_1 e^{-2t}\cos(t) + c_2 e^{-2t}\sin (t) + t e^{-2t} + \frac{1}{2}t e^{-2t}\sin (t)$$

« Last Edit: February 14, 2013, 05:07:49 AM by Victor Ivrii »

#### Marcia Bianchi

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##### Re: TT1--Problem 3
« Reply #2 on: February 14, 2013, 12:50:41 AM »
my solution

#### Victor Ivrii

• Elder Member
• Posts: 2599
• Karma: 0
##### Re: TT1--Problem 3
« Reply #3 on: February 14, 2013, 05:15:11 AM »
Devin, for me WolframAlpha gave exactly your answer for homogeneous equation. Probably you asked it not politely enough

PS Usage of double dollars in LaTeX is deprecated. There is command \Re but out of the box it returns $\mathfrak{R}$ so I redefined it
Code: [Select]
$\renewcommand{\Re}{\operatorname{Re}}$
(dollars needed to tell MathJax to pay attention, in normal LaTeX they would be wrong)

Marcia, I decided that variation of parameters deserves a reward. Note however that you made small computational  errors
« Last Edit: February 14, 2013, 08:41:11 AM by Victor Ivrii »

#### Devin Jeanpierre

• Jr. Member
• Posts: 8
• Karma: 2
##### Re: TT1--Problem 3
« Reply #4 on: February 14, 2013, 07:34:58 AM »
for me, wolfram alpha spits out $y(t) = c_1 e^{-2 t} \sin(t)+c_2 e^{-2 t} \cos(t)+\frac{1}{2} e^{-2 t} (t (\sin(t)+2)+\cos(t))$. It's technically correct, because you can just change the constants to turn it into my solution, but...

PS Usage of double dollars in LaTeX is deprecated.
For  most everything what I really wanted was to use "align*". Turns out mathjax supports this, so I'll be more latexy in the future.