Solve the third-order initial value problem below using the method of Laplace transforms y'''+3y''−18y'−40y=−120 y(0)=6 y'(0)=45 y"(0)=-21

If $xy+8e^y=8e$ , find the value of y" at the point where $x=0$
$y"=?$

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

Find the Laplace transform of the given function
$\{\begin{array}{ll}t& 0,4\le t<\mathrm{\infty }\\ 0& 4\le t<\mathrm{\infty }\end{array}$
$L\{f(t)\}-?$

find the Laplace transform of f (t).
${\displaystyle f\left(t\right)=\frac{e^{3t}-1}{1}}$

Find the Laplace transform of each of the following function:
a. ${\displaystyle \psi \left(t\right)={\int }_0^t{\left(2(t-\tau )\right)}^2\sin \left(4\tau \right)d\tau }$
$L\{\psi (t)\}(s)=?$
b. ${\displaystyle \psi \left(t\right)={\int }_0^t{e}^{-7(t-\tau )}\cos \left(7\tau \right)d\tau }$
$L\{\psi (t)\}(s)=?$

calculate the inverse laplace of ${\displaystyle F\left(s\right)=e^{-3s}\frac{1+s}{s^3+2s^2+2s}}$

Find Laplace Transform of each following
(a) ${\displaystyle t^n}$ , (b) \(\displaystyle{\cos{