Solving a piecewise function by Laplace transform

**Ragnarok** · October 29th 2012, 04:45 PM

Hello, I am having great difficulty with the following problem:

$y''+6y'+5y=g(t)$
where $g(t)=t$ for $0<t<2$ and $g(t)=0$ for $t>2$ . Initial values are $y(0)=1$ and $y'(0)=0$ .

I wrote $g(t)$ as $t-t\cdot U(t-2)$ and used Laplace transforms to get

$(s^2+6s+5)\mathcal{L}\{y\}-s-6=\frac{1}{s}-e^{-2s} \left(\frac{1}{s^2}+\frac{2}{s} \right)$ .

Originally I distributed the $e^{-2s}$ , solved for $\mathcal{L}\{y\}$ and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.

**Prove It** · October 29th 2012, 04:52 PM

Originally Posted by Ragnarok

Hello, I am having great difficulty with the following problem:

$y''+6y'+5y=g(t)$
where $g(t)=t$ for $0<t<2$ and $g(t)=0$ for $t>2$ . Initial values are $y(0)=1$ and $y'(0)=0$ .

I wrote $g(t)$ as $t-t\cdot U(t-2)$ and used Laplace transforms to get

$(s^2+6s+5)\mathcal{L}\{y\}-s-6=\frac{1}{s}-e^{-2s} \left(\frac{1}{s^2}+\frac{2}{s} \right)$ .

Originally I distributed the $e^{-2s}$ , solved for $\mathcal{L}\{y\}$ and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.

It might help you to try to solve the two following DEs...

$\displaystyle \begin{align*} y'' + 6y' + 5y = t \end{align*}$ and $\displaystyle \begin{align*} y'' + 6y' + 5y = 0 \end{align*}$ .

**Ragnarok** · October 29th 2012, 06:56 PM

Okay, I get the solutions

$y=\frac{-6}{25}+\frac{1}{5}t-\frac{13}{50}e^{-5t}+\frac{3}{2}e^{-t}$

for the first one and

$y=\frac{5}{4}e^{-t}-\frac{1}{4}e^{-5t}$

for the second.

These are similar in form to the solution I got in my original attempt at the problem, though I had crazy coefficients. I just don't understand what to do next as I think we're supposed to end up with a function of the form

$f(t-a)\mathclass{U}(t-a)$

and then rewrite it in piecewise form.

**Ragnarok** · October 30th 2012, 09:26 AM

I just solved this problem with the help of my teacher. Thanks!

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