Use properties of the Laplace transform to answer the following (a) If f(t)=(t+5)^2+t^2e^{5t}, find the Laplace transform,L[f(t)] = F(s). (b) If f(t)

A-stability of Heun method for ODEs
I'm trying to determine the stability region of the Heun method for ODEs by using the equation ${\displaystyle y\prime =ky}$, where k is a complex number.

solving second order linear ODE
${\displaystyle \frac{a}{x}{y}^{\prime }\left(x\right)+\frac{1}{2}y{}^{″}\left(x\right)=0}$
where a is a constant. Also how can I solve this equation if the right hand side is different than zero?

How to deal with two interdependent integrators?
I have two functions, f(t,x) and g(t,u), where ${\displaystyle \frac{d}{dt}u=f(t,x)\text{and}\frac{d}{dt}x=g(t,u)}$.
I am trying to discretize the integral of this system in order to track x and u. I have succeeded using Euler integration, which is quite simple, since x(t) and u(t) are both known at t:
${\displaystyle u(t+h)=u\left(t\right)+hf(t,x\left(t\right))}$
${\displaystyle x(t+h)=x\left(t\right)+hg(t,u\left(t\right))}$
However, I am now trying to implement mid-point integration to get more accurate results. (Eventually Runge-Kutta but I am stuck here for now.)

Find the orthogonal trajectory.
${\displaystyle y^2=k{x}^3}$

${\displaystyle y\frac{dy}{dx}=e^x}$
Solve it by separating variables like this:
${\displaystyle ydy=e^{x}dx}$
${\displaystyle \int ydy=\int e^{x}dx}$
${\displaystyle \frac{y^2}{2}=e^x+c}$

Determine $L^{-1}\left[\frac{(s-4)e^{-3s}}{s^2-4s+5}\right]$

I need to calculate a taylor polynomial for a function ${\displaystyle f:R\to R}$ where we know the following
${\displaystyle f{}^{″}\left(x\right)+f\left(x\right)=e^{-x}\mathrm{\forall }x}$
${\displaystyle f\left(0\right)=0}$
${\displaystyle f^{\prime }\left(0\right)=2}$