Question

Using Heun’s Method, compute the value of y at x= 10 taking h= 0.1, Given :

dy/dx = x³ - 2x² + x - 10 and y (0) = 3

ans= f(10)=1783.55 . (using Matlab and show the code)

Answer 1

The student needs to compute the value of y at x=10 using Heun's Method. This can be done programmatically using MATLAB by defining the function for the derivative, initial conditions, step size, and implementing Heun's Method iteratively to approximate the value of y. The provided code snippet illustrates how to achieve this in MATLAB.

Step-by-step explanation:

The student is asking how to compute the value of y at x=10 using Heun's Method with a step size h=0.1, given the differential equation dy/dx = x³ - 2x² + x - 10 and the initial condition y(0) = 3. To solve this using MATLAB, the following code can be used:

function y = heuns_method(dydx, x0, y0, h, xf)
% Number of steps
n = (xf - x0)/h;
% Initialization
y = y0;
% Heun's Method Iteration
for i = 1:n
k1 = dydx(x0, y);
k2 = dydx(x0 + h, y + h*k1);
y = y + (h/2)*(k1 + k2);
x0 = x0 + h;
end
end

% Define the differential equation
f = (x, y) x^3 - 2*x^2 + x - 10;
% Initial condition
x0 = 0;
y0 = 3;
% Step size
h = 0.1;
% Final value of x
xf = 10;

% Call Heun's method
y_final = heuns_method(f, x0, y0, h, xf);
disp(y_final);
When the above code is executed in MATLAB, it will output the value of y at x=10.