Solve System of Differential Equations with Expert Help

find the Laplace transform of the given function.
$f(t)=\{\begin{array}{ll}t,& 0\le t<1\\ 2-t,& 1\le t<2\\ 0,& 2\le t<\mathrm{\infty }\end{array}$

For ${\displaystyle f\left(t\right)=e^{-t}-1}$ , find the Laplace transform of $\left\{\frac{f(t)}{t}\right\}$. Then , for ${\displaystyle g\left(t\right)=e^{-t}-2}$ , examine if the Laplace transform of $\left\{\frac{g(t)}{t}\right\}$ exists.

The Laplace transform of $L\left\{\frac{d^{3}x(t)}{d{t}^3}\right\}$ is given by
Select one:
$sX(s)-x(0)$
${\displaystyle s^{3}X\left(s\right)-s^{2}x\left(0\right)-s\dot{x}\left(0\right)-\ddot{x}\left(0\right)}$
${\displaystyle s^{3}X\left(s\right)-s^2\ddot{x}\left(0\right)-s\dot{x}\left(0\right)-x\left(0\right)}$
${\displaystyle s^{2}X\left(s\right)-sx\left(0\right)-\dot{x}\left(0\right)}$

use laplace transform to solve the following initial value problem:
${\displaystyle 5y^{\prime }+\frac{3}{x}y=3y^2\ln x}$

Find Y(s), the Laplace transform of the below initial value problem's solution, y(t).
$y"+5y=2t^2-9,y(0)=0,y^{\prime }(0)=-3$

Solve the initial value problem below using the method of Laplace transforms.
${\displaystyle y8y^{\prime }+41y=29e^{2t}}$
$y(0)=1,y^{\prime }(0)=7$

Prove that $L^{-1}\left(\frac{s^2}{s^4+4a^4}\right)=\frac{1}{2a}(\cos (h)at\sin at+\sin (h)at\cos at)$

find the Laplace transform of the following periodic functions
$f(t)=\{\begin{array}{ll}t& 0<t<\pi \\ 0& \pi <t<2\pi \end{array}$
Ans, ${\displaystyle F\left(s\right)=\frac{\frac{1}{s^2}(1-e^{-\pi s})-\frac{\pi }{s}{e}^{-\pi s}}{(1-e^{2\pi s})}}$

Find the solution of ${\displaystyle f\left(x\right)=8x+\sin x}$