Riley hobson bence · Problem 5.35

Partial differentiation

Mathematics — Methods · greens-functions-integral-equations

Problem

The function

G(t, \xi)

is defined for

0 \le t \le \pi

G(t, \xi) = \begin{cases} -\cos t \sin \xi & \text{for } \xi \le t, \\ -\sin t \cos \xi & \text{for } \xi > t. \end{cases}

Show that the function

x(t)

defined by

x(t) = \int_0^\pi G(t, \xi) f(\xi)\, d\xi

satisfies the equation

\frac{d^2 x}{dt^2} + x = f(t),

where

f(t)

can be any arbitrary (continuous) function. Show further that

x(0) = [dx/dt]_{t=\pi} = 0

, again for any

f(t)

, but that the value of

x(\pi)

does depend upon the form of

f(t)

. [The function

G(t, \xi)

is an example of a Green's function, an important concept in the solution of differential equations and one studied extensively in later chapters.]