Modified Euler Method to solve initial value problem y'(t)=1/(1+2*t^2),y(0)=1

Introduction: In this question, we apply modified Euler Method to solve initial value problem y'(t)=1/(1+2*t^2),y(0)=1

Question : Apply modified Euler Method to solve initial value problem

\[y’\left(t\right)=\frac{1}{(1+2*t^{2})}, y(0)=1\]

for \(0\le t \le 5 \) with stepsize h =0.5 and also find analytical solution and compare with numerical solutions and graph the solutions.

Solution :

\[y’\left(t\right)=\frac{1}{(1+2*t^{2})}, y(0)=1\]

let \(f(t, y) =\frac{1}{(1+2*t^{2})}, t_0=0, y_0 =1\)

analytical solution of the IVP is

\[y(t) =\sqrt{2}\frac{tan^{-1}(\sqrt{2}t)}{2}+1\]

Modified Euler method

\[y_{i+1}=y_{i}+\frac{h}{2}\left(f(t_{i},y_{i})+f(t_{i+1},y_{i}+h*f(t_{i},y_{i}))\right), i=0,12,3,….,n\]

where \(t_{i+1}=t_i+h\)

stepsize \(h =0.5\)

to find \(y(5)\) we