Part A-

Convert the equation-

d³y/dx³ - 2 · dy/dx - 1 · x · y = 0

and initial conditions y(0) = 1.00279, y'(0) = 1.13313 and y''(0) = -0.17902, into three first order differential equations. Find the formulas for solving this system of equations by:

1. Euler's forward difference method

2. The second order Runge-Kutta method based on

k₁ = h·f(x_n, v_n)

k₂ = h·f(x_n + h/2, y_n + k₁/2)

y_n+1 = y_n + k₂

Find the numerical solution y at 0.1, x = 0.2 and x = 0.3 by both methods using a step length h = 0.1.

Give your answers to 5 decimal places (no more and no less) however you should do you calculations with sufficient accuracy to ensure you answers are accurate to 5 decimal places.

For the Euler method:

y₀ = 1.00279

y₁ =

y₂ =

y₃ =

For the Runge-Kutta method:

y₀ = 1.00279

y₁ =

y₂ =

y₃ =

Part B-

Find the numerical solution to

y''(x) - (x + 0.2)y'(x) - (x + 3.7)y(x) = 0.5x + 3.6

in the interval 0 ≤ x ≤ 1 with conditions y(0) = 2.5895 and y(1) = 2.9327, using a step length h = 0.2 and the 2nd-order Runge-Kutta scheme given by

k₁ = h·f(x_n, y_n)

 k₂ = h·f(x_n + 2h/3, y_n + 2k₁/3)

y_n+1 = y_n + k₁/4 + 3k₂/4

Give your answers to 5 decimal places (no more and no less).

y₀ = 2.5895

y₁ =

y₂ =

y₃ =

y₄ =

y₅ =2.932