1. ## Range-Kutta

Equation:
$\displaystyle I \theta '' + c \theta ' = Tm$
I = 0.9
c = 0.15
Tm = 0.3

My solution:
Reduced Equation 1° order:
$\displaystyle y_2' = \frac{T_m}{I} - \frac{Cy_2}{I}$

I used three steps to RK2 and found:
$\displaystyle y_1 = 0.01665$
$\displaystyle y_2 = 0.03306$
$\displaystyle y_3 = 0.04249$

Is that correct?
How do I find the analytical solution?

2. I don't know about the Range-Kutta part but as far as an analytical solution goes you've already taken the first step yourself.

$\displaystyle y' = \frac{T_m}{I} - \frac{C}{I}y$

This is a Linear First Order differential equation. You can, if you've ever covered linear first order ODE's, solve this for $\displaystyle y = \theta'$ and then go from there.

3. It seems to me the OP is looking for advice on how to solve the DE using Runge-Kutta not any other way.

Follow this example Runge Kutta Methods

4. Originally Posted by Apprentice123
Equation:
$\displaystyle I \theta '' + c \theta ' = Tm$
I = 0.9
c = 0.15
Tm = 0.3

My solution:
Reduced Equation 1° order:
$\displaystyle y_2' = \frac{T_m}{I} - \frac{Cy_2}{I}$

I used three steps to RK2 and found:
$\displaystyle y_1 = 0.01665$
$\displaystyle y_2 = 0.03306$
$\displaystyle y_3 = 0.04249$

Is that correct?
What are the initial conditions and what the step size? Also are you not supposed to be finding $\displaystyle$$\theta$ ?

CB

5. Originally Posted by Apprentice123
Equation:
$\displaystyle I \theta '' + c \theta ' = Tm$
I = 0.9
c = 0.15
Tm = 0.3

How do I find the analytical solution?
This is an inhomogeneous linear constant coefficient ODE. You should have covered how to solve this.

The general solution is the sum of the general solution of the homogeneous equation

$\displaystyle I \theta '' + c \theta ' = 0$

and a particular integral of the original equation, and a PI for this is $\displaystyle T_m t$.

To get the actual solution you now apply the initial conditions to the general solution.

CB

6. The initial conditions are
$\displaystyle y_0 = 0$ and $\displaystyle x_0 = 0$

How to find the local and global errors?

7. Originally Posted by Apprentice123
The initial conditions are
$\displaystyle y_0 = 0$ and $\displaystyle x_0 = 0$

How to find the local and global errors?
Are you not supposed to be finding $\displaystyle \theta(x)$ ? If so the initial conditions will specify $\displaystyle \theta '(0)$ and $\displaystyle \theta(0)$

CB

8. Originally Posted by Apprentice123
Equation:
$\displaystyle I \theta '' + c \theta ' = Tm$
I = 0.9
c = 0.15
Tm = 0.3

My solution:
Reduced Equation 1° order:
$\displaystyle y_2' = \frac{T_m}{I} - \frac{Cy_2}{I}$
You reduce the second order ODE to a first order system as shown in the second post in this thread: http://www.mathhelpforum.com/math-he...ta-162591.html

CB