Use Eulers method with step size 03 to estimate y15 where yx

Use Euler\'s method with step size 0.3 to estimate y(1.5), where y(x) is the solution of the initial-value problem y\' = -2x + y^2, y(0) = 1.

Solution

Given differential equation : y\' = -2x + y^2,

f(x, y ) = -2x + y^2

and also given y(x) = y(0) = 1

x₀ = 0 , y₀ =1

f₀ = f( 0, 1) = -2(0) + (1)^2

f₀ = 1

And given step size h = 0.3

y₁ = y₀ + h f(x_{0 ,} y₀ )

= 1 + 0.3 (1 )

y₁ = 1.3

for next step , since h = 0.3 , the next point is x+ h = 0 + 0.3 = 0.3 ,we substitute what we know in euler\'s formula , and we have

y(x + h) = y(x) + h f(x, y )

y₂ = y₁ + h f(x1 , y₁)

y₂ = 1.3 + 0.3( -2 *0.3 + (1.3)^2)

= 1.3 + 0.3 (-0.6 + 1.69)

y₂ = 1.627

now for third step ; x2 = 0.3 + 0.3 = 0.6 , y2 = 1.627

y3 = y2 + h f(x2 , y2 )

y3 = 1.627 + 0.3 (- 2 * 0.6 + (1.627)^2)

= 1.627 + 0.3 ( -1.2 + 2.6471)

= 2.061

y4 = y3 + h f( x3, y3) ; x3 = 0.6 + 0.3= 0.9

= 2.061 + 0.3 (- 2 * 0.9 + (2.061)^2)

= 2.061 + 0.3 ( -1.8 + 4.2477)

y4 = 2.795

y5 = y4 + h f(x4 , y4) , x4 = 0.9+0.3 = 1.2

= 2.795 + 0.3 ( -2 * 1.2 + (2.795)^2))

= 2.795 + 0.3 ( -2.4 + 7.812)

y5 = 4.4186

y6 = y5 + h f( x5 , y5) ; x5 = 1.2 +0.3 = 1.5

= 4.418 + 0.3 ( -2 * 1.5 + (4.418)^2)

= 4.418 + 0.3( -3 + 19.518)

= 9.3736

therefore y( 1.5 ) = 9.3736