Okay, I know this is alot... but I am stuck, so here goes...(adsbygoogle = window.adsbygoogle || []).push({});

Use the method of Laplace transform to solve the initial value problem

[tex]y''+3ty'-6y=0, y(0) = 1, y'(0) = 0[/tex]

[tex]L\{y'' + 3ty' - 6y\} = L\{0\}[/tex]

[tex]s^{2}Y(s) - sy(0) - y'(0) + 3L\{ty'\} - 6Y(s) = 0[/tex]

[tex]s^{2}Y(s) - s(1) - 0 - \frac{d}{ds}\left(3 L\{ty'\}\right) -6Y(s) = 0[/tex]

Now to resolve the [tex]- \frac{d}{ds}\left(3 L\{ty'\}\right)[/tex]

[tex]= - \frac{d}{ds}\left(3 L\{ty'\}\right)[/tex]

[tex]= - \frac{d}{ds}3 \left(sY(s) - y(0)\right)[/tex]

[tex]= -3sY'(s) - 3Y(s)[/tex]

Plugging it back into the eq we now have

[tex]s^{2}Y(s) - s - 3sY'(s) - 3Y(s) - 6Y(s) = 0[/tex]

[tex]-3sY'(s) + (s^{2}-9)Y(s) - s = 0[/tex]

[tex]Y'(s) + \left(-\frac{s}{3} + \frac{3}{s}\right)Y(s) = -\frac{1}{3}[/tex]

[tex]\mu = e^{\int\left(-\frac{s}{3} + \frac{3}{s}\right)ds}[/tex]

[tex]\mu = e^{\left(-\frac{s^{2}}{6} + ln(s^{3})\right)[/tex]

[tex]\mu = s^{3} e^{\left(-\frac{s^{2}}{6}\right)[/tex]

[tex]\int\left(\frac{d}{ds}(s^{3}e^{-\left(\frac{s^2}{6}\right)}Y(s)\right) = \int-\left(\frac{1}{3}\right)s^{3}e^{-\left(\frac{s^2}{6}\right)} ds[/tex]

[tex]s^{3}e^{-\left(\frac{s^2}{6}\right)}Y(s) = \int-\left(\frac{1}{3}\right)s^{3}e^{-\left(\frac{s^2}{6}\right)} ds[/tex]

RIGHT SIDE

[tex]=\left(\frac{1}{3}\right)(-3(s^{2}+6)e^{-\left(\frac{s^2}{6}\right)[/tex]

[tex]=(s^2+6)e^{-\left(\frac{s^2}{6}\right) + A[/tex]

[tex]Y(s)=\frac{(s^2+6)}{s^{3}} + \frac{A e^ \frac{s^2{6}{s^{3}}[/tex]

[tex] Limit...as... s \rightarrow \infty........Y(s) = 0...therefore A = 0[/tex]

[tex]Y(s) = \frac{s^2+6}{s^3}[/tex]

Break down the Inverse Laplace

[tex]L^{-1}\{\frac{s^2+6}{s^3}\}[/tex]

[tex]=L^{-1}\{\frac{s^2}{s^3}\} + L^{-1}{\frac{6}{s^3}\}[/tex]

[tex]=L^{-1}\{\frac{1}{s}\} + L^{-1}{\frac{6}{s^3}\}[/tex]

[tex]= 1 + ?????? [/tex]

This is where I get lost.... I don't know how to do the other side... Please help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Laplace Transofrm using IVP

**Physics Forums | Science Articles, Homework Help, Discussion**