To solve the given second-order differential equation in the general form:
[
\frac{d^2x}{dt^2} + f(x) = F(x) \sin(Mt),
]
we will use the method of multiple scales and averaging techniques, which are well-suited for systems with fast and slow time scales. The solution will be expressed as:
[
x(t) = x_0(t) + h X(t),
]
where:
- ( x_0(t) ) is the solution when ( F = 0 ),
- ( h ) is a small parameter,
- ( M ) is much larger than the natural frequency of ( x_0(t) ),
- ( X(t) ) and ( \sin(Mt) ) are fast-varying components,
- ( x(t) ), ( f(x) ), and ( F(x) ) are slow-varying components.
Step 1: Solve for ( x_0(t) ) (Unperturbed Solution)
When ( F = 0 ), the equation reduces to:
[
\frac{d^2x_0}{dt^2} + f(x_0) = 0.
]
This is a nonlinear oscillator equation, and its solution depends on the form of ( f(x_0) ). For simplicity, assume ( f(x_0) ) is linear, i.e., ( f(x_0) = \omega_0^2 x_0 ), where ( \omega_0 ) is the natural frequency. The solution is:
[
x_0(t) = A \cos(\omega_0 t + \phi),
]
where ( A ) and ( \phi ) are constants determined by initial conditions.
Step 2: Introduce the Perturbation
Now, consider the full equation with ( F \neq 0 ). Assume ( h ) is a small parameter, and write the solution as:
[
x(t) = x_0(t) + h X(t),
]
where ( X(t) ) captures the fast-varying components due to the ( \sin(Mt) ) term. Substitute this into the original equation:
[
\frac{d^2}{dt^2}(x_0 + h X) + f(x_0 + h X) = F(x_0 + h X) \sin(Mt).
]
Expand ( f(x_0 + h X) ) and ( F(x_0 + h X) ) using a Taylor series around ( x_0 ):
[
f(x_0 + h X) \approx f(x_0) + h X f'(x_0) + \mathcal{O}(h^2),
]
[
F(x_0 + h X) \approx F(x_0) + h X F'(x_0) + \mathcal{O}(h^2).
]
Substitute these into the equation and collect terms of order ( h^0 ) and ( h^1 ):
-
Order ( h^0 ):
[
\frac{d^2x_0}{dt^2} + f(x_0) = 0,
]
which is satisfied by ( x_0(t) ).
-
Order ( h^1 ):
[
\frac{d^2X}{dt^2} + f'(x_0) X = F(x_0) \sin(Mt).
]
Step 3: Solve for ( X(t) ) (Fast-Varying Component)
The equation for ( X(t) ) is a forced linear oscillator with a time-dependent coefficient ( f'(x_0) ). Since ( M \gg \omega_0 ), the forcing term ( F(x_0) \sin(Mt) ) drives the system at a high frequency. Assume ( X(t) ) has the form:
[
X(t) = B(t) \sin(Mt) + C(t) \cos(Mt),
]
where ( B(t) ) and ( C(t) ) are slowly varying amplitudes. Substitute this into the equation and use the method of averaging to separate fast and slow dynamics. After averaging over the fast time scale, the equations for ( B(t) ) and ( C(t) ) are:
[
\frac{dB}{dt} \approx -\frac{F(x_0)}{2M} \cos(\omega_0 t + \phi),
]
[
\frac{dC}{dt} \approx \frac{F(x_0)}{2M} \sin(\omega_0 t + \phi).
]
Integrate these to find ( B(t) ) and ( C(t) ).
Step 4: Combine Solutions
The final solution is:
[
x(t) = x_0(t) + h X(t),
]
where:
- ( x_0(t) = A \cos(\omega_0 t + \phi) ),
- ( X(t) = B(t) \sin(Mt) + C(t) \cos(Mt) ).
This solution captures both the slow dynamics of ( x_0(t) ) and the fast oscillations driven by ( \sin(Mt) ).
Step 5: Validate the Solution
The solution is valid under the assumptions:
- ( h ) is small,
- ( M \gg \omega_0 ),
- ( f(x) ) and ( F(x) ) are slowly varying compared to ( \sin(Mt) ).
For further details on the method of multiple scales and averaging, refer to (Nayfeh, 2005) and (Sanders, 1982).