Direction Fields and Numerical Methods

Question

Use Euler's method with step size 0.4 to estimate y(0.8), where y(x) is the solution of the initial-value problem y' = -3X-2 +y^2, y(0) = -1 what is y (0.8)?

Answer 1

dy/dx = f(x,y) = -3x-2 +y^2

x0 = 0, y0 = -1

y1=y0+f(x0,y0)(x1-x0)

x1 = 0.4, y1= -1+f(0,-1)(0.4-0) = -1+(-1)(0.4-0) = -1.4

x2 = 0.8, y2=y1+f(x1,y1)(x2-x1) = -1.4 + f(0.4, -1.4)(0.4) = -1.4+(-1.24)*0.4= -1.896

Answer 2

Just to flesh out more of what the other tutor already answered:

----- 1: Identify your initial conditions -----

We're starting with x = 0 and y = -1.

----- 2: Make the first step -----

The first step is from x = 0 to x = 0.4.

First, calculate f(x, y) = -3x - 2 + y^2. At x = 0 and y = -1, we get:

f(0, -1) = -3(0) - 2 + (-1)^2 = -2 + 1 = -1.

Now update y using the formula y_next = y_current + step_size * f(x, y):

y_1 = -1 + 0.4 * (-1) = -1 - 0.4 = -1.4.

So after the first step, x = 0.4 and y = -1.4.

----- 3: Make the next step -----

Our next step is from x = 0.4 to x = 0.8.

Calculate f(x, y) again. At x = 0.4 and y = -1.4, we get:

f(0.4, -1.4) = -3(0.4) - 2 + (-1.4)^2 = -1.2 - 2 + 1.96 = -1.24.

Now update y:

y_2 = -1.4 + 0.4 * (-1.24) = -1.4 - 0.496 = -1.896.

----- Final result -----

At x = 0.8, y is approximately -1.896.

Remember, for each step we need to just recalculate f(x, y). If you needed to keep making steps, you could actually iterate through part 3 of this process.

Cheers!