All these questions must be answered using Python:

Question $1$

$(1)$ Write a Euler's method code in Python to solve $y^{\text{'}}=x$ y for $y(1)=1$ on the interval $1,2.$

$(2)$ Compare your numerical solution to the exact solution of the equation.

$(3)$ What is the maximum absolute value of the error?

Question $2$

$(1)$ Repeat question $1$ for $y^{\text{'}\text{'}}+y^{\text{'}}-y=0,y(0)=-1,y^{\text{'}}(0)=1.$

Question $3$

$(1)$ Instead of Euler's method in Question $1,$ use a fourth Runge$-$Kutta method

Question $4$

$(1)$ Repeat question $2$ using a fourth$-$order Runge$-$Kutta method. $.$

Homework:$-$

Compare Euler, RK$4,$ two step Adams Bashforth and the two step Adams$-$Moulton method to

solve

$(1)y^{\text{'}}=xy,y(0)=1,x$ in $0,5$