An initial-value problem is given by the differential equation,
f(x,y) = x + y, y(0) = 1.64
The Euler-midpoint method is used to find an approximate value to y(0.1) with a step size of h = 0.1.
Then use the integrating factor method, to find the exact value of y(0.1).
Hence, determine the global error, giving your answer to 5 decimal places.
got stuck with this. anybody can help?
To solve this initial-value problem using the Euler-midpoint method, we start by approximating the value of y(0.1) by taking step sizes of h = 0.1.
The Euler-midpoint method involves two steps:
1. Using the initial condition, calculate the approximate value of y at x = 0.1 by taking a midpoint evaluation.
2. Using the calculated midpoint value, calculate the approximate value of y at x = 0.1 using the slope calculated at the midpoint.
Let's break down the steps:
Step 1:
Start with the initial condition y(0) = 1.64.
Since we need to find the value of y at x = 0.1, we take a step of h from x = 0 to x = 0.1.
Calculate the midpoint value using the formula:
k1 = h * f(0, y(0)) = 0.1 * f(0, 1.64) = 0.1 * (0 + 1.64) = 0.164
Step 2:
Using the midpoint value, calculate the slope at the midpoint (0.05, 1.82) as follows:
k2 = h * f(0.05, y(0.05)) = 0.1 * f(0.05, 1.82) = 0.1 * (0.05 + 1.82) = 0.188
Now, we can calculate the approximate value of y at x = 0.1 using the formula:
y(0.1) ≈ y(0) + k2 = 1.64 + 0.188 = 1.828
Therefore, using the Euler-midpoint method, the approximate value of y(0.1) is 1.828.
Now, let's use the integrating factor method to find the exact value of y(0.1).
The integrating factor method involves rewriting the given differential equation in the form:
dy/dx + P(x)y = Q(x),
where the integrating factor, call it u, is given by:
u(x) = e^(∫P(x)dx).
In our case, the differential equation is:
dy/dx = x + y.
To rewrite it in the desired form, we subtract y from both sides:
dy/dx - y = x.
Comparing this with the general form, we have:
P(x) = -1 and Q(x) = x.
To find the integrating factor, we calculate:
u(x) = e^(∫-1dx) = e^(-x).
Now, we multiply both sides of the rewritten equation by the integrating factor u(x):
e^(-x)dy/dx - e^(-x)y = xe^(-x).
The left-hand side simplifies to:
d/dx(ye^(-x)) = xe^(-x).
Integrating both sides with respect to x, we get:
ye^(-x) = ∫(xe^(-x))dx.
Integrating using any integration technique, we find:
ye^(-x) = -xe^(-x) - e^(-x) + C,
where C is the constant of integration.
To find the exact value of y(0.1), we substitute x = 0.1:
y(0.1)e^(-0.1) = -0.1e^(-0.1) - e^(-0.1) + C.
Simplifying, we find:
y(0.1) = -0.1 - e^(0.1) + Ce^(0.1).
The constant C can be determined by substituting the initial condition y(0) = 1.64:
1.64 = -0.1 - 1 + Ce^(0).
Simplifying, we find:
C = 2.74.
Substituting this value of C back into the equation for y(0.1), we get:
y(0.1) = -0.1 - e^(0.1) + 2.74e^(0.1).
Calculating this expression, we find that the exact value of y(0.1) is approximately 1.833.
Finally, to find the global error, which measures the difference between the exact and approximate values, we subtract the approximate value from the exact value:
Global error = Exact value - Approximate value = 1.833 - 1.828 = 0.005.
Therefore, the global error, to 5 decimal places, is 0.005.