hi

I need your help to resolve this question

y''-y'-2y=18e^-t sin3t

y(c)=0 , y'(c)=3

?

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- Oct 5th 2012, 10:33 AMlooloQuestion on the applications of Laplace transforms
hi

I need your help to resolve this question

y''-y'-2y=18e^-t sin3t

y(c)=0 , y'(c)=3

? - Oct 5th 2012, 10:47 AMTheEmptySetRe: Question on the applications of Laplace transforms
- Oct 5th 2012, 11:32 AMlooloRe: Question on the applications of Laplace transforms
ammmmmm

I want a solution by this law

Attachment 25057 - Oct 5th 2012, 12:32 PMTheEmptySetRe: Question on the applications of Laplace transforms
I don't see how that rule applies to this problem. Since we don't know the intial contidtions at zero we assume

$\displaystyle y(0)=c_1 \quad y'(0)=c_2$

If we take the laplace transfrom of the equation we get

$\displaystyle s^2Y-sc_1-c_2-sY+c_1-2Y=\frac{54}{(s+1)^2+9}$

Solving for Y gives

$\displaystyle Y=\frac{c_2-c_1}{(s-2)(s+1)}+\frac{c_1s}{(s-2)(s+1)}+\frac{54}{(s-2)(s+1)[(s+1)^2+9]}$

Edit: You could use the theorem to invert the 2nd of the two terms in this problem. - Oct 5th 2012, 01:06 PMlooloRe: Question on the applications of Laplace transforms
i Began resolve the question as follows

Attachment 25062 - Oct 5th 2012, 01:46 PMTheEmptySetRe: Question on the applications of Laplace transforms
- Oct 5th 2012, 02:23 PMlooloRe: Question on the applications of Laplace transforms
ammmmmm

I'll try to find a way s-axis translation theorem.(Smirk) - Oct 5th 2012, 05:01 PMProve ItRe: Question on the applications of Laplace transforms
Are you able to use Partial Fractions to simplify before trying to take the Inverse transform to get back to y? If not, you may need to use the convolution theorem...

- Oct 6th 2012, 01:30 AMlooloRe: Question on the applications of Laplace transforms
Ammmmmm

Dr. asked us to use partial fractions

But I do not know how - Oct 6th 2012, 02:05 AMlooloRe: Question on the applications of Laplace transforms
I'm so sorry,

the Conditions in

y(0)=0 ,

y(0)=3

this is my Try...

Attachment 25073 - Oct 6th 2012, 03:28 AMlooloRe: Question on the applications of Laplace transforms
Now I want to use partial fractions..

Can you help me ؟

Attachment 25074 - Oct 6th 2012, 05:53 AMProve ItRe: Question on the applications of Laplace transforms
First of all you have an arithmetic error. When you add 3 to the other side and make a common denominator, it should be $\displaystyle \displaystyle \begin{align*} \frac{3\left[ (s + 1)^2 + 9 \right]}{(s + 1)^2 + 9} \end{align*}$, which when you expand it out gives $\displaystyle \displaystyle \begin{align*} \frac{3(s + 1)^2 + 27}{(s + 1)^2 + 9} \end{align*}$. As for the rest, you need to solve for $\displaystyle \displaystyle \begin{align*} Y(s) \end{align*}$ then take the inverse transform to get back to $\displaystyle \displaystyle \begin{align*} y(t) \end{align*}$.

- Oct 6th 2012, 04:50 PMlooloRe: Question on the applications of Laplace transforms
ammmmmmm

Can you help me complete the answer - Oct 6th 2012, 06:52 PMMaxJasperRe: Question on the applications of Laplace transforms
In the diff eq, change t to t+c and take LaplaceT and solve for y(t).

$\displaystyle \mathcal{L}_t[y(t)](s)$=$\displaystyle \frac{3 e^{-c} \left(e^c s^2+2 e^c s+6 s \sin (3 c)+10 e^c+6 \sin (3 c)+18 \cos (3 c)\right)}{\left(s^2-s-2\right) \left(s^2+2 s+10\right)}$

and an inverse LapT would look like:

$\displaystyle y(t)=e^{-3 c-t} \left(e^{c+3 t}+e^{3 t} \sin (3 c)-e^{3 c} \sin (3 t)+e^{3 t} \cos (3 c)+e^{3 c} \cos (3 t)-e^{4 c}-2 e^{3 c} \cos (3 c)\right)$ - Oct 6th 2012, 06:58 PMlooloRe: Question on the applications of Laplace transforms