Find the Laplace transform of \(\displaystyle{f{{\left({t}\right)}}}={10}{t}{e}^{{-{5}{t}}}\)

solve the given initial value problem
${\displaystyle y{}^{‴}+3y{}^{″}-13y^{\prime }-15y=0,y\left(0\right)=y^{\prime }\left(0\right)=0,y{}^{″}\left(0\right)=1}$

find the given inverse Laplace transform
$L^{-1}\left\{\frac{s^2+1}{s(s-1)(s+1)(s-2)}\right\}$

Topic:
Exact Differential Equations
Instructions:
Solve the Exact Differential Equations of GIven Problem #2 only, ignore the other given problem.
Note:
Please answer all of the problems in the attached photo. I will rate you with “like/upvote” after; if you answer all of the problems. If not, I will rate you “unlike/downvote.” Kindly show the complete step-by-step solution.
1. ${\displaystyle (x{e}^{x}y+e^{x}y)dx+(8+x{e}^x)dy=0;y\left(0\right)=-4}$
2. ${\displaystyle \left(\frac{x+y}{y^2+1}\right)dy+(\arctan y+x)dx=0}$

Find the inverse laplace transforms for the following functions
$F(s)=\frac{e^{-3s}}{s^2}$
$F(s)=\frac{2e^{-5s}}{s^2+4}$

Find the general solution of the given differential equations
${\displaystyle 4y\prime \prime -4y\prime -3y=0}$

Help with a generating function and differential equation
I have a generating function that Im

use properties of Laplace transform