Question

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' = x2y − 1 2 y2, y(0) = 9. (Round your answer to four decimal places.) y(1) =

Answer 1

Answer: -1.0018*10^42

Step-by-step explanation:

The Euler Method in numerical analysis is used to approximate the solution to an initial value problem using the tangent line to the solution curve through the (x0,y0) to obtain such approximations.

The euler method equation is:

Yn+1 = Yn + h*f(Xn,Yn)

Where n = number of steps, h = Xn+1 - Xn, f(Xn,Yn) is the slope of the curve at (Xn,Yn).

Variables given in the equation

Y(X=0) = 9,

X0 = 0.

h = step = 0.2

f(X,Y) = X^2*Y - 12*Y^2

B

For the first step, n = 0 in the euler equation. Therefore we have:

Y1 = Y0 + h*f(X0,Y0)

Substituting the Y0 = 9 and X0 = 0 into the function f(X,Y) = X^2*Y - 12*Y^2 = 0^2*9 - 12*9^2 = -972.

Therefore Y1 = 9 + 0.2*(-972) = -185.4.

For n = 1

Repeating the same step with X1 = X0 + h = 0 + 0.2 = 0.2,

Y1 = -185.4,

h = 0.2.

Substitute the variables into the equation Y2 = Y1 + h*f(X1,Y1) and f(X1,Y1) = X1^2*Y1 - 12*Y1^2

Y2 = -82682.4672.

Continue the iterations following the steps above till the result is reached.

The summary of the iteration.

When n = 0, X = 0, Y = -185.4

When n = 1 , X = 0.2, Y = -82682.4672

When n = 2 , X = 0.4, Y = -1.6407*10^10

When n = 3, X = 0.6, Y = -6.4609*10^20

When n = 4, X = 0.8, Y = -1.0018*10^42