x’ = 2x + 3y , x(0) = -2.7
y’ = 2x + y , y(0) = 2.8
[ 0 , 1 ] h = 0.05
EULER, RK4 AND TRAPEZOIDAL.. FIRST 10 ITERATION ONLY
To solve the given system of differential equations using Euler's method, RK4 method, and Trapezoidal method, we will iterate through the equations for a specified number of iterations.
Euler's method approximates the solution by using the following formula:
x(i+1) = x(i) + h * f(x(i), y(i))
y(i+1) = y(i) + h * g(x(i), y(i))
where f(x, y) and g(x, y) are the given functions in the system, h is the step size, and x(i) and y(i) represent the approximated values at the ith iteration.
RK4 method approximates the solution using the following formulas:
k1x = h * f(x(i), y(i))
k1y = h * g(x(i), y(i))
k2x = h * f(x(i) + h/2, y(i) + k1y/2)
k2y = h * g(x(i) + h/2, y(i) + k1y/2)
k3x = h * f(x(i) + h/2, y(i) + k2y/2)
k3y = h * g(x(i) + h/2, y(i) + k2y/2)
k4x = h * f(x(i) + h, y(i) + k3y)
k4y = h * g(x(i) + h, y(i) + k3y)
x(i+1) = x(i) + (k1x + 2*(k2x + k3x) + k4x)/6
y(i+1) = y(i) + (k1y + 2*(k2y + k3y) + k4y)/6
Trapezoidal method approximates the solution using the following formulas:
x_temp = x(i) + h * f(x(i), y(i))
y_temp = y(i) + h * g(x(i), y(i))
x(i+1) = x(i) + h/2 * (f(x(i), y(i)) + f(x_temp, y_temp))
y(i+1) = y(i) + h/2 * (g(x(i), y(i)) + g(x_temp, y_temp))
Now, let's compute the first 10 iterations for each method using the given system of differential equations and the provided initial values:
Euler's method:
Initialize x(0) = -2.7 and y(0) = 2.8, and h = 0.05.
1. Compute x(1) and y(1):
x(1) = -2.7 + 0.05 * f(-2.7, 2.8)
y(1) = 2.8 + 0.05 * g(-2.7, 2.8)
2. Repeat step 1 for the next iterations until i = 10.
RK4 method:
Initialize x(0) = -2.7 and y(0) = 2.8, and h = 0.05.
1. Compute x(1) and y(1) using the RK4 formulas.
2. Compute x(2) and y(2) using the RK4 formulas, using the updated values from the previous step.
3. Repeat steps 1 and 2 until i = 10.
Trapezoidal method:
Initialize x(0) = -2.7 and y(0) = 2.8, and h = 0.05.
1. Compute x_temp and y_temp using the Trapezoidal formulas.
2. Compute x(1) and y(1) using the updated values from the previous step.
3. Repeat steps 1 and 2 until i = 10.