Numerical solution for 2nd order ODE using Euler Method

Hello, i am trying to solve y''+4*y'+4y=t^3*e^(2*t) with initial values of y'(0) = 0 and y(0) = 0 using Euler's method to get the numeric solution but i don't know how to start

Answer 1

[Sam Chak]

Hi @KG

If you show us the Euler method formula, then we don't have to search in Google or Wikipedia. This also ensures that the formula you give to us is correct and reliable with source cited. Anyhow, here is the demo. Hope that this is the Euler solution that you are looking for and acceptable.

function Demo_Euler

close all;

clc

tStart = 0;

step = 1e-2; % if step is smaller, then numerical solution is more accurate

tEnd = 1;

t = [tStart:step:tEnd]; % simulation time span

y0 = [0, 0]; % initial values

f = @(t,x) [x(2); % system of 1st-order ODEs

(t^3)*exp(2*t) - 4*x(1) - 4*x(2)];

yEuler = EulerSolver(f, t, y0); % calling Euler Solver

% analytical solution (can be obtained by hand, or by using 'dsolve')

yExact = (1/128)*exp(-2*t).*(3*(1 + t) + exp(4*t).*(-3 + 9*t - 12*t.^2 + 8*t.^3));

% compare numerical solution with analytical solution

plot(t, yEuler(1,:), t, yExact)

grid on

xlabel('Time, t')

ylabel('y_{1} and y_{2}')

title('Time responses of the system')

legend('Euler solution', 'Exact solution', 'location', 'best')

end

function y = EulerSolver(f, x, y0)

y(:, 1) = y0; % initial condition

h = x(2) - x(1); % step size

n = length(x); % number of steps

for i = 1:n-1

y(:, i+1) = y(:, i) + h*f(x(i), y(:, i));

end

end

Result:

Answer 2

[SUBRATA]

clc

close all

clear all

%All the function

f=@(t,y,z)z;

g=@(t,y,z)(t^3*(exp(2*t))-4*z-4*y);

t0=input('input of the initial t_0=');

y0=input('input of the initial y_0=');

z0=input('input of the initial z_0=');

tn=input('input of the t_n=');

h=input('step size=');

n=(tn-t0)/h;

t(1)=t0;y(1)=y0;z(1)=z0;

fprintf(' time \t\t y\t\t z \n');

for i=1:n

y(i+1)=y(i)+h*f(t(i),y(i),z(i));

z(i+1)=y(i)+h*g(t(i),y(i),z(i));

t(i+1)=t(i)+h;

fprintf(' %3.4f \t\t\t %3.4f \t\t\t %3.4f \n',t(i+1),y(i+1),z(i+1));

plot(t,y)

end