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)=
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To use Euler's method to estimate the value of y(1) for the given initial-value problem, we will follow these steps:
Step 1: Determine the number of subintervals.
Since the step size is given as 0.2 and we need to find y(1) which is 1 unit away from the initial point x=0, we'll have 1/0.2 = 5 subintervals.
Step 2: Define the initial values.
Given that y(0) = -1, this will be our initial condition.
Step 3: Set up the iteration.
We will use Euler's method to compute the values of y for each subinterval. The formula is as follows:
y(n+1) = y(n) + h * f(x(n), y(n))
where:
- y(n+1) is the approximate value of y at the next subinterval.
- y(n) is the approximate value of y at the current subinterval.
- h is the step size (0.2 in this case).
- f(x(n), y(n)) is the derivative of y with respect to x evaluated at the current subinterval.
Step 4: Compute the values.
Using the given initial values and Euler's method formula, we can compute the approximate values of y at each subinterval:
First subinterval (n=0):
x(0) = 0
y(0) = -1
f(x(0), y(0)) = 5(0) + (-1)^2 = 1
y(1) = -1 + 0.2 * 1 = -0.8
Second subinterval (n=1):
x(1) = 0.2
y(1) = -0.8 (from the previous subinterval)
f(x(1), y(1)) = 5(0.2) + (-0.8)^2 = 1.16
y(2) = -0.8 + 0.2 * 1.16 = -0.568
Third subinterval (n=2):
x(2) = 0.4
y(2) = -0.568 (from the previous subinterval)
f(x(2), y(2)) = 5(0.4) + (-0.568)^2 = 1.101216
y(3) = -0.568 + 0.2 * 1.101216 = -0.3477672
Fourth subinterval (n=3):
x(3) = 0.6
y(3) = -0.3477672 (from the previous subinterval)
f(x(3), y(3)) = 5(0.6) + (-0.3477672)^2 = 1.00975559
y(4) = -0.3477672 + 0.2 * 1.00975559 = -0.145816482
Fifth subinterval (n=4):
x(4) = 0.8
y(4) = -0.145816482 (from the previous subinterval)
f(x(4), y(4)) = 5(0.8) + (-0.145816482)^2 = 0.798714406
y(5) = -0.145816482 + 0.2 * 0.798714406 = 0.4616227076
Step 5: Estimate y(1).
Since the fifth subinterval corresponds to x=1, the value of y(1) will be the approximate value we obtained at the fifth subinterval:
y(1) ≈ 0.4616227076.
Therefore, using Euler's method with a step size of 0.2, we estimate that y(1) ≈ 0.4616227076.