This Must Be Done Using the Program MATLAB In class we considered the equation y

Question

This Must Be Done Using the Program MATLAB

In class we considered the equation y = y2 3x, where y(0) = 1.

a. Use an Euler approximation with a step size of 0.1 to approximate y(2).

b. Use a Runge-Kutta approximation with a step size of 0.5 to approximate y(2).

c. Graph both approximation functions in the same window as a slope field for the differential equation.

d. Find a formula for the actual solution (not by hand!) and compute its value at x = 2. (You can use vpa(...) to compute the result as a decimal expression.) Is the explicit formula useful? Is the computed value of y(2) useful? In your comments, explain

Explanation / Answer

clear all

f = @(x,y) (y^2 - 3*x);

x = 0;

i = 1;

y(1) = 1;

for i = 2: 21

y(i) = y(i-1) + 0.1*(f( x(i-1), y(i-1)));

x(i) = x(i-1) + 0.1;

end

y(i)

disp('rk4')

%rk4%

z(1) = 1;

for i = 2:21

k1 = f(x(i-1), z(i-1));

k2 = f(x(i-1) + 0.05, z(i-1) + 0.05*k1);

k3 = f(x(i-1) + 0.05, z(i-1) + 0.05*k2);

k4 = f(x(i-1) + 0.1, z(i-1) + 0.1*k3);

x(i) = x(i-1) + 0.1;

z(i) = z(i-1) + 0.1/6*(k1 + 2*k2 + 2*k3+ k4);

end

z(i)

y is the euler approximation at 2 and z(i) is the rk4 approximatiom for 2.

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