Compute the first five iterations of eulers method for the g

Compute the first five iterations of euler\'s method for the given IVP along with the error at each step by completing the (partially filled) table below, using h = 0.2. Round all your values to four places after the decimal.

Solution

given y\' = -ty , y(0) = 1, h = 0.2

f(x, y) = -ty

the intial condition tells the value of intial coordinates is y_{0 =} 1

Now we use the Euler method formulas to generate values for y₁

The Y iteration formula with n = 0 gives us:

y₁ = y₀ + h f(y₀)

y_{1 =} 1+ 0.2 * (t₀ * y₀ )

= 1 + 0.2 * (0.0 * 1 )

= 1

y₂ = y₁ + h f(y₁)

= 1 +0.2 * (-t1 * y1)

= 1 + 0.2(-0.2 * 1)

= 1 - 0.04 = 0.96

n t_n   y_n y(t_n) Error = y_n - y(t_n)

0 0.0 1 0 1

1 0.2 1 -0.2 1.2

2 0.4 0.96 -0.384 1.344

3 0.6 0.883 -0.529 1.412

4 0.8 0.777 -0.621 1.398

5 1.0 0.653 -0.653 1.306