Diff eqn IVP
posted by shasha .
The differential equation that governs the forced oscillation is shown below:
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

Normalize 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)+(24A6B)sin(4t)
Compare coefficients of cos(4t) and sin(4t):
24A6B=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.
Respond to this Question
Similar Questions

Forced Oscillation  drwls => I need your help =)
A 2.00kg object attatched to a spring moves without friction and is driven by an external force given by F= (3.00N)sin(2pi*t) The force constant of the spring is 20N/m. Determine a) period b) amplitude of motion a)T= 2pi/omega T= 2pi/ … 
Maths
Solve the differential equation: d2y/dx2  2 dy/dx + y = 3sinhx The answer should be: y(x) = e^x (Ax+ B) + ( 3^8 )(2[x^2][e^x]  [e^x] ) Can someone please show me how to work it out? 
Undamped, forced oscillation
Undamped, forced oscillation?!?!? I have this question and can't seem to get the right answer. Damping is negligible for a 0.130 kg object hanging from a light 6.30 N/m spring. A sinusoidal force with an amplitude of 1.70 N drives 
Diff eqn IVP
A particle moves on the xaxis with an acceleration, a=(6t4)ms⁻1. Find the position and velocity of the particle at t=3 , if the particle is at origin and has a velocity of when t=0 
Diff eqn IVP
A simple pendulum of length is oscillating through a small angle θ in a medium for which the resistance is proportional to the velocity. Obtain the differential equation of its motion and discuss the motion. 
MATH
Differential equations, initial value problem. The general equation of motion is: mx"+Bx'+kx=f(t), where the independent variable is t, and the displacement x is the dependent variable. In this case, external force f(t)=0, so mx"+Bx'+kx=0 … 
Math
An external sinusoidal force is applied to an oscillating system which can be modelled by a model spring and a damper. The general solution of the equation of the motion is given by, x = 5 + 3.071cos(2πt – β) + 3.14exp(7.96t) … 
differential equation
Write the model ODE and the implied initial values, and then solve the IVP to answer each of the f ollowing: [2*8=16pts] (a) A 3kg mass is attached to a spring whose constant is 6 N/m, and the entire system is submerged in a liquid … 
Physics
Explain what is meant by i) free oscillation ii) damped oscillation iii) forced oscillation iv) resonance. 
Differential Equations
Verify whether the given equation(y=c1cos(x+c2)) is a solution to this equation[(d^2y/dx^2)^2 + (dy/dx)^2  1 = 0].