# 3rd order differential equation

1. Jul 18, 2012

### dominicfhk

1. The problem statement, all variables and given/known data
I took differential eq-n class more than a year ago and I am having problem solving this problem.
5(d^3y/dt^3)+3(d^2y/dt^2)+7(dy/dt)+y = 4(d^2f/dt^2), where f(t)=cos(t)-2sin(t)

2. Relevant equations
none

3. The attempt at a solution
First thing I did was to take the second derivative of f(t) and multiply it by 4 for the right side of the equation, which turns out to be 4(-cos(t)+2sin(t))=-4cos(t)+8sin(t).
For the left hand side, I take its characteristic equation 5y^3+3y^2+7y+1=0 and tried to solve for the roots of the cubic equation, but turns out the roots are not whole & real numbers:
t1 = -0.15009708847612485
t2 = 0.22495145576193756+1.1321959748355932i
t3 = 0.22495145576193756-1.1321959748355932i
Then I am not sure how to proceed... or did I use a wrong approach? If not then how should I proceed the problem with what method? (undetermined coefficients?) Not sure if this helps, this differential equation describes a circuit with y(t) as the output and f(t) as the input. I was asked to find the steady-state output of the circuit. So I figure I should take the limit of the solution of the differential equation as t approach infinity.

Any help will be appreciated!

2. Jul 18, 2012

### ehild

In the stationary state, the output will be of the same frequency as the input signal. The real part of all roots are negative, you made a mistake when copying them. The solutions of the homogeneous equation tend to zero with time. Assume the stationary solution in form of Acos(t)+Bsin(t) and solve for A, B.

ehild