Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

Here I have one task and it is preparation for small exam. I solved it by hand for first case 1), but I need to check it in $Mathematica$ and to try to implement it for both cases 1) and 2) automatically. After decoupling I am looking for solutions of the decoupled system and how to get them and come back to the source variables because I need solutions of them?

System of two differential equations of 4th order and I need to transform them in 1) equivalent system of 4 equations of second order with appropriate substitutions and 2) 8 equations of first order. How to do that in Mathematica automatically, in one case 4 equations second order maximum and in another case 8 equations of first order maximum? Example is with known constants a and b. I know that I must use some substitutions to reduce the system.

Firstly can be helpful homogenous solution when g1[x] and g2[x] are equal to zero.

first eq

  g1[x]==a1*X1[x] + a2*X2[x] + a3*Derivative[2][X1][x] - 
  a4*Derivative[2][X2][x] + a5*Derivative[4][X1][x]

second eq

  g2[x]==b2*X1[x] + b1*X2[x] - b4*Derivative[2][X1][x] + 
  b3*Derivative[2][X2][x] + b5*Derivative[4][X2][x]

1) 4 equations with adopted substitutions of second order differential equations.

2) 8 equations of first order

I did a) by hand in the form

$(\text{x1}=\text{X1}(x)) (\text{x3}=\text{X2}(x)) \left(\text{x2}=\text{X1}''(x)\right) \left(\text{x4}=\text{X2}''(x)\right) $

$ X(x)=\left( \begin{array}{c} \text{x1}(x) \\ \text{x2}(x) \\ \text{x3}(x) \\ \text{x4}(x) \\ \end{array} \right) $ $, A=\left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ -\frac{\text{a3}}{\text{a1}} & -\frac{\text{a2}}{\text{a1}} & -\frac{\text{a5}}{\text{a1}} & -\frac{\text{a4}}{\text{a1}} \\ 0 & 0 & 0 & 1 \\ -\frac{\text{b5}}{\text{b1}} & -\frac{\text{b4}}{\text{b1}} & -\frac{\text{b3}}{\text{b1}} & -\frac{\text{b2}}{\text{b1}} \\ \end{array} \right)$ $, B=\left( \begin{array}{c} 0 \\ \frac{\text{g1}(x)}{\text{a1}} \\ 0 \\ \frac{\text{g2}(x)}{\text{b1}} \\ \end{array} \right)$ $$ X''(x)=A X(x)+B$$

share|improve this question
    
@Nasser thank you, it is corrected now. It is not partial, just of one variable –  Pipe Jan 26 '13 at 6:33
    
@Naser it can e helpful also homogenous solution firstly when the functions g1 and g2 are equal to zero –  Pipe Jan 26 '13 at 6:44
    
@I didn't speak about standard state space formulation. I need solutions of system of differential equations and how to transform it in 4 second order –  Pipe Jan 26 '13 at 14:42
add comment

1 Answer

I am a little confused by your terms and what you have shown. But in case it helps, one way to obtain the state space matrix is to use StateSpaceModel as follows

eq1 = g1[x] == 
   a1*X1[x] + a2*X2[x] + a3*Derivative[2][X1][x] - 
    a4*Derivative[2][X2][x] + a5*Derivative[4][X1][x];

eq2 = g2[x] == 
   b2*X1[x] + b1*X2[x] - b4*Derivative[2][X1][x] + 
    b3*Derivative[2][X2][x] + b5*Derivative[4][X2][x];

StateSpaceModel[{eq1, 
  eq2}, {{X1'[x], 0}, {X1[x], 0}, {X2'[x], 0}, {X2[x], 0}}, {{g1[x], 
   0}, {g2[x], 0}}, {X1[x], X2[x]}, x]

Which gives

Mathematica graphics

We see there are 8 state variables as expected.

You find more information on the above command here http://reference.wolfram.com/mathematica/ref/StateSpaceModel.html

share|improve this answer
    
I need 4 second order and solutions. –  Pipe Jan 26 '13 at 14:43
    
what I can do with this? –  Pipe Jan 26 '13 at 14:44
    
vote down sorry, you didnt understand –  Pipe Jan 26 '13 at 17:03
    
@naser what does two separated columns and rows represent? what is a solution of this matrix of differential equations, how to reform now the system in 8 diff equations –  Pipe Jan 26 '13 at 23:32
    
my full description of the problem is here, maybe you can understand what actually I need. Thank you for your attention and try. math.stackexchange.com/questions/287762/… –  Pipe Jan 27 '13 at 2:35
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.