4th order RK to solve 2nd order ODE

[Kanashii]

Homework Statement

Consider the initial value problem x" + x′ t+ 3x = t; x(0) = 1, x′(0) = 2 Convert this problem to a system of two first order equations and determine approximate values of the solution at t=0.5 and t=1.0 using the 4th Order Runge-Kutta Method with h=0.1.

Homework Equations

y_n+1 = y_n + 1/6 ( k₁ + 2k₂ + 2k₃ + k₄)
k₁ = h* f(x_n, y_n)
k₂ = h* f(x_n + 1/2 h, y_n + 1/2 k₁)
k₃ = h* f(x_n + 1/2 h, y_n + 1/2 k₂)
k₄ = h* f(x_n+1, y_n+k₃)

The Attempt at a Solution

I tried to convert the equation into two linear ODEs:

x = x₁
x′= x₁' = x₂
x′′= x₁'' = x₂'

x₂' = -x₂ t + t - 3x₁ (first linear ODE)
x₁' = x₂ (second linear ODE)

I do not know what to do from here.
From 4th Order RK equations, I do not know what f(x_n, y_n) is.

 

[eys_physics]

The Runge-Kutta method give the solution to the differential equation:
$$x'(t)=f(x,t)$$

This can also be generalized to several equations by allowing $x(t)$ and $f(x,t)$ being vectors, i.e. $x=(x_1,x_2,...,x_n)$ and $f=(f_1,f_2,...,f_n)$. So, in your case $f=(f_1,f_2)$ is a two-dimensional array containing the right-hand side of each equation.

By the way, I recommend that you use Latex to write your equations. In your post it is difficult to read them.

 

[Ray Vickson]

Homework Statement

Consider the initial value problem x" + x′ t+ 3x = t; x(0) = 1, x′(0) = 2 Convert this problem to a system of two first order equations and determine approximate values of the solution at t=0.5 and t=1.0 using the 4th Order Runge-Kutta Method with h=0.1.

Homework Equations

y_n+1 = y_n + 1/6 ( k₁ + 2k₂ + 2k₃ + k₄)
k₁ = h* f(x_n, y_n)
k₂ = h* f(x_n + 1/2 h, y_n + 1/2 k₁)
k₃ = h* f(x_n + 1/2 h, y_n + 1/2 k₂)
k₄ = h* f(x_n+1, y_n+k₃)

The Attempt at a Solution

I tried to convert the equation into two linear ODEs:

x = x₁
x′= x_1′ = x₂
x′′= x_1′′ = x_2′

x_2′ = -x₂ t + t - 3x₁ (first linear ODE)
x_1′ = x₂ (second linear ODE)

I do not know what to do from here.
From 4th Order RK equations, I do not know what f(x_n, y_n) is.

See
http://math.stackexchange.com/quest...-order-method-on-a-system-of-2-first-order-od
for a detailed explanation plus worked example.

BTW: I do not agree that your work is difficult to read; in fact, it reads surprisingly well and looks quite good. However, post #2 is correct in suggesting the use of LaTeX; you would find it much easier and faster than constantly using the [ S U B] ... [/ S U B] construction, and it wold look even better than what you have already.

 

[epenguin]

Some of the ' need to be outside of the [ SUB] ' [/SUB]

 

[Mark44]

Some of the ' need to be outside of the [ SUB] ' [/SUB]

I moved them, so it should be OK now.