Solve in Matlab In class we considered the equation y y2 3x

Solve in 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, explai

Solution

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.