# Diff eqn- IVP

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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

• Differential equations-missing one initial condition -

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)+(-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.

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