a Consider a forced oscillator with response yt satisfying S

(a) Consider a forced oscillator with response y(t) satisfying State with reasoning (but before solving), whether or not you expect this system to exhibit pure resonance. (b) Find the general solution of (1). (c) Find the solution of (1) if the system is initially at rest undisplaced , i.e. y(0) = y?(0) = y(0)=0. (d) Give Mathematica commands, about 5 lines, to find the solution y(t) of the IVP comprising (1),(ICI,2). and to plot y(t) for 0

Solution

(a) The system is expected to exhibit resonance because of presence of cos(4t) factor in the equation.

(b) Consider characteristic equation: r²+16=0 => r² = -16 => r=+4i, -4i

So the complimentary solution is given by : y_c = Acos(4t)+Bsin(4t)

Now in order to find particular solution, consider:

=> y_p = t(Acos(4t)+Bsin(4t))

Now y_p\' = Acos(4t)+Bsin(4t)-4tAsin(4t)+4tBcos(4t)

=> y_p\" = -4Asin(4t)+4Bcos(4t)-4Asin(4t)-16tAcos(4t)+4Bcos(4t)-16tBsin(4t) = -8Asin(4t)+8Bcos(4t)-16tAcos(4t)-16tBsin(4t)

So put this in original equation, we get:

-8Asin(4t)+8Bcos(4t)-16tAcos(4t)-16tBsin(4t)+16tAcos(4t)+16tBsin(4t) = 5cos(4t)

Equating like terms, we get following two equations:

-8A-16tB+16tB = 0 => -8A = 0 => A=0

8B-16tA+16tA = 5 => 8B = 5 => B =5/8

So the particular solution is given by:

y_p = 5/8tsin(4t)

Hence the general solution y(t) = y_c + y_p = Acos(4t)+Bsin(4t)+5t/8sin(4t)

(c) Now when y(0) =0 => 0=Acos(0)+Bsin(0)+0 => 0=A

Also y\'(t) = -4Asin(4t)+4Bcos(4t)+5/8sin(4t)+5t/2cos(4t)

Now y\'(0) = 0=> 0 = -4Asin(0)+4Bcos(0)+5/8sin(0)+0 => 0=4B => B=0

So the solution is given by y(t) = 5t/8sin(4t)