Transform differential equations into state space form

[irishetalon00]

Homework Statement

I have derived the differential equations of a system. They are like the following:
$$a\ddot{\theta} - b\ddot{x} + c \theta = 0 \\ d\ddot{\theta} + e\ddot{x} = F(t)$$
where a,b,c,d,e are constants.

I'm having trouble putting it into state space form, since I have the highest derivative in both equations. Can anyone show me how this is done?

Homework Equations

The Attempt at a Solution

 

[MathematicalPhysicist]

You have: $\ddot{\theta}(ae+db)+(ec)\theta = cF(t)$, from this you can pick $u=\theta , v =\dot{\theta}$, another equation is:
$(ea+bd)\ddot{x}-cd\theta=aF(t)$, I don't see how to use this equation; do you have other constraints?

 

[irishetalon00]

Thank you for looking into this. How did you transform into the equations you have listed?

No, I don't have any other constraints, unfortunately.

 

[MathematicalPhysicist]

The RHS of the first equation should be: $bF(t)$, I just multiplied $b$ the second equation and multiplied the first equation by $e$ and added the two equations.Similar operations have been done to make my second equation.

 

[MathematicalPhysicist]

The first ODE I wrote can be solved by multiplying the ode by $\dot{\theta}$ and integrating.