Remove function from differential system d^2x/dt^2 + a*dx/dt = j, dj/dt = -b*j - dx/dt + u(t)

[ChuckNoise]

Hi.

I am a bit stuck with an assignment and i hope that someone can give me a notch in the right direction.

I got a model for a robot arm which is made up of 2 differential equations:

d^2x/dt^2 + a*dx/dt = j (1)
dj/dt = -b*j - dx/dt + u(t) (2)

j = j(t) is the power through the motor x = x(t) is the position of the load, both functions of time. The voltage on the system u = u(t) controls the robot arms movement adx/dt is the mechanical loss in rotor aswell as movement of the arm with a load. bj descripes the electrical resistance in the motors electrical circuit. a and b are positive real constant which is far below 1, meaning that |a-b| < 2.

I have to remove j(t) from the system (1-2) and show that:
d^3x/dt^3 +(a+b)d^2x/dt^2 + (1 + ab)*dx/dt = u

My initial approach was to diffentiate equation 1 which led to:
d^3x/dt^3 + a* d^2x/dt^2 = dj/dt

i then substituted the left side of equation 1 aswell as the j on the right side of equation 2 leading to:
d^3x/dt^3 + a* d^2x/dt^2 = -b(d^2x/dt^2 + a*dx/dt) - dx/dt + u(t)

Since that i have tried a couple of diffentent approaches, but i can't seem to figure it out. Can anyone help me here?
Thanks!

 

[HallsofIvy]

That's a good start! You have d^3x/dt^3 + a* d^2x/dt^2 = -b(d^2x/dt^2 + a*dx/dt) - dx/dt + u(t).
Now do the algebra to d^3x/dt^3+ a d^2x/dt^2= -(bd^2) d^2x/dt^2- (ab) dx/dt- dx/dy+ u(t).

d^3x/dt^3+ (a+ bd^2)d^2x/dt^2+ (ab- 1)dy/dt= u(t).

That's a "linear third order ordinary differential equation with constant coefficients". Its solution depends on solving its characteristic equation r^3+ (a+ bd^2)r^2+ (ab-1)r= 0. An obvious first step is to factor out 'r': r(r^2+ (a+ bd^2)r+ ab-1)= 0. So one solution is r= 0 (which means that x(t)= a constant) and you can use the quadratic formula to determine the other.