what is the laplace transformation of the function in the picture f(x)=xsin x

Second derivative using implicit differentiation with respect to x of ${\displaystyle x=\sin y+\cos y}$
${\displaystyle 1=\cos y\frac{dy}{dx}-\sin y\frac{dy}{dx}}$
${\displaystyle 1=\frac{dy}{dx}(\cos y-\sin y)}$
${\displaystyle \frac{dy}{dx}=\frac{1}{\cos y-\sin y}}$

Solve by Laplace transformation method the following D.E.
(1) $y^{″}-3y^{\prime }+2y=4^2{e}^t$ given that $y(0)=-3,y^{\prime }(0)=5$​​​​​​​

A thermometer is taken from an inside room to the outside ,where the air temperature is ${\displaystyle 5^{\circ }F}$. After 1 minute the thermometer reads ${\displaystyle {55}^{\circ }F}$, and after 5 minutes it reads ${\displaystyle {30}^{\circ }F}$, what is the initial temperature of the inside room?

Solve for G.S. / P.S. for the following differential equations using the method of solution for homogeneous equations.
${\displaystyle -(3x+25y)dx+(25x+3y)dy=0}$

Reverse Laplace transform of
${\displaystyle \frac{3s-15}{2s^2-4s+10}}$

Use the Laplace Transform to solve the given intitial value problem.
${\displaystyle y=\sqrt{2}\sin \left(\sqrt{2}t\right),y\left(0\right)=7,y^{\prime }\left(0\right)=2}$

Find the general solution of the given differential equation. ${\displaystyle y”+y^{\prime }-6y=12e^{3t}+12e^{-2t}}$