Page 36 - Essentials of applied mathematics for scientists and engineers

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26 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
2.1.24 Lessons
When the differential equation is nonhomogeneous use the linearity of the differential equation
to transfer the nonhomogeneous condition to one of the boundary conditions. Usually this will
result in a homogeneous partial differential equation and an ordinary differential equation.
We pause here to note that while the method of separation of variables is straightforward
in principle, a certain amount of intuition or, if you wish, cleverness is often required in order
to put the equation and boundary conditions in an appropriate form. The student working
diligently will soon develop these skills.

Problems
1. Using these ideas obtain a series solution to the boundary value problem

u t = u xx
u(t, 1) = 0
u(t, 0) = 0
u(0, x) = 1

2. Find a series solution to the boundary value problem
u t = u xx + x
u x (t, 0) = 0

u(t, 1) = 0
u(0, x) = 0

2.2 VIBRATIONS
In vibrations problems the dependent variable occurs in the differential equation as a second-
order derivative of the independent variable t. The methodology is, however, essentially the same
as it is in the diffusion equation. We first apply separation of variables, then use the boundary
conditions to obtain eigenfunctions and eigenvalues, and use the linearity and orthogonality
principles and the single nonhomogeneous condition to obtain a series solution. Once again, if
there are more than one nonhomogeneous condition we use the linear superposition principle
to obtain solutions for each nonhomogeneous condition and add the resulting solutions. We
illustrate these ideas with several examples.

Example 2.5. A Vibrating String

Consider a string of length L fixed at the ends. The string is initially held in a fixed position
y(0, x) = f (x), where it is clear that f (x)mustbezeroatboth x = 0and x = L. The boundary