Monday

September 1, 2014

September 1, 2014

Posted by **shasha** on Monday, March 14, 2011 at 7:36am.

0.2 d²y/dt² + 1.2 dy/dt +2y = r(t) where r(t) is the external force.

Given that r(t) = 5 cos 4t with y(0) = 0 . find the equation of motion of the forced oscillations

- Differential equations-missing one initial condition -
**MathMate**, Monday, March 14, 2011 at 8:57amNormalize the equation by multiplying by 5:

0.2 d²y/dt² + 1.2 dy/dt +2y = 5 cos(4t) = r(t)

to:

d²y/dt² + 6 dy/dt + 10y = 25cos(4t)

Find the complementary solution:

m²+6m+10=0

m=-3±i

So the solution to the homogeneous equation is:

yc=e^(-3t)(C1*cos(t)+C2*sin(t))

Now find the particular solution by undetermined coefficients:*Assume*the particular solution to be:

yp=Acos(4t)+Bsin(4t)

and substitute in y of the the original equation:

d²yp/dt² + 6 dyp/dt + 10yp = 25cos(4t)

-16Acos(4t)-16Bsin(4t)

+6(4Bcos(4t)-4Asin(4t))

+10Acos(4t)+10Bsin(4t)

=(-6A+24B)cos(4t)+(-24A-6B)sin(4t)

Compare coefficients of cos(4t) and sin(4t):

-24A-6B=0 => B=-4A

-6A+24B=25 => -102A=25 => A=-25/102

Therefore

yp(t)=-(25/102)cos(4t)+(100/102)sin(4t)

(substitute in homogeneous equation to verify that you get 25cos(4t) )

The general solution is therefore:

y=yc+yp=e^(-3t)(C1*cos(t)+C2*sin(t))-(25/102)cos(4t)+(100/102)sin(4t)

Initial conditions:

To solve the second order problem completely, you'll need two initial conditions. We are givn y(0)=0 at t=0.

We need another one (such as y'(0)=5 at t=0).

Substitute the initial conditions into the general solution above and solve for C1 and C2 to give the final solution of the initial value problem.

**Answer this Question**

**Related Questions**

Diff eqn- IVP - A simple pendulum of length is oscillating through a small ...

Diff eqn- IVP - A particle moves on the x-axis with an acceleration, a=(6t-4)ms...

Undamped, forced oscillation - Undamped, forced oscillation?!?!? I have this ...

Forced Oscillation - drwls => I need your help =) - A 2.00kg object attatched...

Math - For the harmonic potential V(x,y) = x^2 + y^2 a) Find the total ...

Physics - For the harmonic potential V(x,y) = x^2 + y^2 a)Find the total ...

math - The displacement s (in metres) of a body in a damped mechanical system, ...

algebra 1,urgent - the expression (4x^2-3x+1)-(x^3+2x+7) what is the coefficient...

math - Please help me on this calculus question. I am getting the wrong diff. ...

Physics - 2.0 kg weight is on simple, horizontally oscillation (in a spring) ...