(Solved): 4). a) Derive the formula of Euler's Method to find the digitized solution to y(x)=f(x,y) and th ...

4). a) Derive the formula of Euler's Method to find the digitized solution to $y^{?}(x)=f(x,y)$ and the error formula for Euler's Method. Solve $y^{?}(x)=3y+3x$ over $[0,0.3]$ with initial condition $y(0)=1$ and $\mathrm{h}=0.1$. b) Solve the ODE in part (a) by using Runge Kutta Method of order 2. c) Compare with $y(x)=\frac{4}{3}{e}^{3x}?x?\frac{1}{3}$ and calculate true error and approximate error for $x=0.1$.