1. Week 10 6. Eigenvalue problems for ODEs Example 1:
Consider a boundary-value problem for a 2nd-order linear homogeneous ODE,
(1)
(2)
where λ is a constant.
(1)-(2) are obviously satisfied if y(x) = 0 for all x, which is called the trivial solution.
It turns out that (1)-(2) may also have non-trivial solutions, but this occurs only for special values of λ.

2. These values and the corresponding non-trivial solutions are called the eigenvalues (EVs) and the eigenfunctions (EFs), and (1)-(2) is an eigenvalue problem (EVP).
To solve (1)-(2), first solve equation (1)...
(3)
Substitute (3) into the boundary conditions (2):
= 1
= 0
≠ 0
Observe that B can’t be zero (why?) – hence...

3. hence,
(4)
where n = ±1, ±2, ±3... is a non-zero (why?) integer.
The EFs can be found by substituting A = 0 and equality (4) into the general solution (3),
(5)
Comments: Typical features of eigenvalue problems
Note that EVP (1)-(2) has infinitely many solutions.
Note that the constant B in expression (5) for the EFs remains undetermined.

4. Example 2:
Solve the following EVP:
where λ is the eigenvalue.
Example 3:
Solve the following EVP:

5. 6. Modelling: Forced oscillations
Let an elastic spring be attached to the wall and a body of mass m:
Let the body’s coordinate x be counted from the equilibrium position (such that the spring is neither squeezed nor stretched).
The body’s motion is thus fully described by the function x(t), where t is the time (don’t get confused by the unusual notation).

6. Note that
Newton’s Second Law states that
hence,
(6)
where Fexternal represents human intervention, wind, earthquake, gravity from Alpha Centauri, etc.
Regarding Fspring, the physicists told us that...

7. (Hook’s Law)
where k is the spring’s modulus (to be measured by the physicists). The “–” shows that the spring force is of opposite direction to the displacement.
Regarding Ffriction , the physicists told us that
(7)
where the “–” shows that the friction force is opposite to the velocity.
(7) can be rewritten in the form

8. The most interesting case is that of a periodic external forcing – so let’s make it, for simplicity, sinusoidal:
where F0 is the amplitude and Ω is the frequency of the forcing (if Ω is large, Fexternal changes direction very frequently).
Now, equation (6) becomes
(8)
where
۞ ω is called the natural frequency (of an oscillating system).

9. First, consider the unforced undamped case (A = γ = 0), so that (8) becomes
The general solution of this ODE can be easily found:
and it describes oscillations of constant amplitude.
If the initial displacement and velocity are both zero, i.e.
(9)
the solution is x = 0 for all t, i.e. the oscillator remains at rest forever.

10. Next, consider the forced undamped case (A ≠ 0, γ = 0), so that (8) becomes
This ODE can be solved using the MoUC (the basic rule)...
Given initial conditions (9) of zero displacement and velocity, one readily obtains
Observe that this solution exists only if Ω ≠ ω.

11. If Ω = ω (i.e. if the frequency of the forcing equals the oscillating system’s natural frequency), we have to use the modification rule of the MoUC. For the initial conditions (9), we obtain
This solution can be separated into the oscillatory part and the amplitude:
One can see that the amplitude grows linearly with time – which physically corresponds to a phenomenon called resonance.

12. In manufacturing and architecture, resonances are very dangerous and should be avoided.
There are two famous cases where they were not:
Friction, generally, causes resonance to saturate.
If, for example, we let γ ≠ 0 in equation (8), the solution will initially grow, but eventually stabilise. In terms of the MoUC, we’ll be back to the basic rule in this case.
However, the amplitude at which oscillations saturate can be (and sometime is) too large.
A more effective way of getting rid of a resonance is “detuning”.