Use Euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem.
y' = 5x + y^2, y(0)=-1.
y(1)=
To use Euler's method, we'll start by approximating the solution of the differential equation at each step, using the given step size of 0.2.
First, let's define the differential equation:
dy/dx = 5x + y^2
And the initial condition:
y(0) = -1
Now, we'll proceed step by step to estimate y(1):
Step 1:
First, calculate the slope at the initial point (0, -1):
slope = 5(0) + (-1)^2 = 1
Now, approximate the value of y(0.2) using the Euler's method:
y(0.2) ≈ y(0) + (slope)(step size)
y(0.2) ≈ -1 + (1)(0.2)
y(0.2) ≈ -0.8
Step 2:
Calculate the slope at the point (0.2, -0.8):
slope = 5(0.2) + (-0.8)^2 = 0.86
Approximate the value of y(0.4) using the Euler's method:
y(0.4) ≈ y(0.2) + (slope)(step size)
y(0.4) ≈ -0.8 + (0.86)(0.2)
y(0.4) ≈ -0.728
Step 3:
Calculate the slope at the point (0.4, -0.728):
slope = 5(0.4) + (-0.728)^2 = 0.86
Approximate the value of y(0.6):
y(0.6) ≈ y(0.4) + (slope)(step size)
y(0.6) ≈ -0.728 + (0.86)(0.2)
y(0.6) ≈ -0.646
Step 4:
Calculate the slope at the point (0.6, -0.646):
slope = 5(0.6) + (-0.646)^2 = 0.993
Approximate the value of y(0.8):
y(0.8) ≈ y(0.6) + (slope)(step size)
y(0.8) ≈ -0.646 + (0.993)(0.2)
y(0.8) ≈ -0.447
Step 5 (Final step):
Calculate the slope at the point (0.8, -0.447):
slope = 5(0.8) + (-0.447)^2 = 3.82
Approximate the value of y(1):
y(1) ≈ y(0.8) + (slope)(step size)
y(1) ≈ -0.447 + (3.82)(0.2)
y(1) ≈ 0.798
Therefore, using Euler's method with a step size of 0.2, the estimated value of y(1) is approximately 0.798.
To use Euler's method with a step size of 0.2 to estimate y(1), we first need to calculate the number of steps required. Since our step size is 0.2 and we want to estimate y(1), we have a total of (1-0)/0.2 = 5 steps.
Now, let's calculate the estimate of y(1) using Euler's method. We'll start with the initial condition y(0) = -1 and iteratively calculate the next values.
Step 1:
Given x0 = 0 and y0 = -1, we can calculate the slope at this point:
f0 = 5x0 + y0^2 = 5(0) + (-1)^2 = 1
Next, we can calculate the next value of y using Euler's method:
y1 = y0 + h * f0 = -1 + 0.2 * 1 = -0.8
Step 2:
Given x1 = 0.2 and y1 = -0.8, we can calculate the slope at this point:
f1 = 5x1 + y1^2 = 5(0.2) + (-0.8)^2 = 4.36
Next, we can calculate the next value of y using Euler's method:
y2 = y1 + h * f1 = -0.8 + 0.2 * 4.36 = 0.872
Step 3:
Given x2 = 0.4 and y2 = 0.872, we can calculate the slope at this point:
f2 = 5x2 + y2^2 = 5(0.4) + (0.872)^2 = 3.35
Next, we can calculate the next value of y using Euler's method:
y3 = y2 + h * f2 = 0.872 + 0.2 * 3.35 = 1.042
Step 4:
Given x3 = 0.6 and y3 = 1.042, we can calculate the slope at this point:
f3 = 5x3 + y3^2 = 5(0.6) + (1.042)^2 = 4.368
Next, we can calculate the next value of y using Euler's method:
y4 = y3 + h * f3 = 1.042 + 0.2 * 4.368 = 1.916
Step 5:
Given x4 = 0.8 and y4 = 1.916, we can calculate the slope at this point:
f4 = 5x4 + y4^2 = 5(0.8) + (1.916)^2 = 7.449
Next, we can calculate the next value of y using Euler's method:
y5 = y4 + h * f4 = 1.916 + 0.2 * 7.449 = 3.406
Finally, we have estimated the value of y(1) using Euler's method with a step size of 0.2 as y(1) = 3.406.