Final answer:
The amplitude of the periodic solution to the given equation is 4/(4 + c^2). As the parameter c increases, the amplitude decreases. For c = 2, the general solution of the equation is given by y(t) = Aexp(-t/2)cos(√7t/2) + Bexp(-t/2)sin(√7t/2), and all solutions tend to the periodic solution y(t) = (4/5)cos(t) + (8/5)sin(t) as t approaches positive infinity.
Step-by-step explanation:
To find a periodic solution of the given equation, we can substitute the given form of the solution, y(t) = acost + bsint into the differential equation y'' = -3y - cy' + sint. After substituting, we get:
-a^2sint - b^2sint - cbsint + 3acost - 2bsint = sint
Comparing the coefficients of sint and cost on both sides, we can obtain the following equations:
-a^2 - b^2 - cb = 1
3a - 2b = 0
Solving these equations will give us the values of a and b, which represent the amplitude of the solution:
a = 4/(4 + c^2)
b = (2c)/(4 + c^2)
Therefore, the amplitude of the solution is 4/(4 + c^2).
To show that the amplitude of the solution decreases as the parameter c increases, we can observe that as c increases, the value of the denominator in the amplitude formula (4 + c^2) also increases. Since the numerator (4) remains constant, the amplitude decreases.
For c = 2, the general solution of the differential equation is y(t) = Aexp(-t/2)cos(√7t/2) + Bexp(-t/2)sin(√7t/2), where A and B are constants determined by initial conditions. This general solution contains an exponential decay term that causes the solutions to approach zero as t tends to infinity. As a result, all solutions tend to the periodic solution y(t) = (4/5)cos(t) + (8/5)sin(t) as t approaches positive infinity.