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.