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Determine the Green's function for the boundary value problem

y'' + y = 0

with...

Jul 2, 2024

Determine the Green's function for the boundary value problem

y'' + y = 0

with

y(0) = 0

and

y(pi) = 0

Solution by Steps

step 1

To determine the Green's function for the boundary value problem

y'' + y = 0

with

y(0) = 0

and

y(\pi) = 0

, we first solve the homogeneous equation

y'' + y = 0

step 2

The general solution to the homogeneous equation

y'' + y = 0

y(x) = A \sin(x) + B \cos(x)

step 3

Apply the boundary conditions to find

A

and

B

. For

y(0) = 0

, we have

A \sin(0) + B \cos(0) = 0 \implies B = 0

step 4

With

B = 0

, the solution simplifies to

y(x) = A \sin(x)

. Apply the second boundary condition

y(\pi) = 0

A \sin(\pi) = 0 \implies A = 0

step 5

Since both

A

and

B

are zero, the homogeneous solution is

y(x) = 0

. Now, we construct the Green's function

G(x, \xi)

for the inhomogeneous problem

step 6

The Green's function

G(x, \xi)

must satisfy

G''(x, \xi) + G(x, \xi) = \delta(x - \xi)

with boundary conditions

G(0, \xi) = 0

and

G(\pi, \xi) = 0

step 7

For

x \neq \xi

, the Green's function satisfies the homogeneous equation

G''(x, \xi) + G(x, \xi) = 0

. Thus,

G(x, \xi)

can be written as

G(x, \xi) = \begin{cases} A(\xi) \sin(x) + B(\xi) \cos(x) &amp; \text{for } x &lt; \xi \\ C(\xi) \sin(x) + D(\xi) \cos(x) &amp; \text{for } x &gt; \xi \end{cases}

step 8

Apply the boundary conditions:

G(0, \xi) = 0 \implies B(\xi) = 0

and

G(\pi, \xi) = 0 \implies C(\xi) \sin(\pi) + D(\xi) \cos(\pi) = 0 \implies D(\xi) = 0

step 9

The Green's function simplifies to

G(x, \xi) = \begin{cases} A(\xi) \sin(x) &amp; \text{for } x &lt; \xi \\ C(\xi) \sin(x) &amp; \text{for } x &gt; \xi \end{cases}

step 10

Ensure continuity at

x = \xi

A(\xi) \sin(\xi) = C(\xi) \sin(\xi) \implies A(\xi) = C(\xi)

step 11

To satisfy the jump condition at

x = \xi

, we integrate the differential equation around

\xi

\int_{\xi-\epsilon}^{\xi+\epsilon} (G''(x, \xi) + G(x, \xi)) \, dx = \int_{\xi-\epsilon}^{\xi+\epsilon} \delta(x - \xi) \, dx \implies [G'(x, \xi)]_{\xi-\epsilon}^{\xi+\epsilon} = 1

step 12

This gives

G'(\xi^+, \xi) - G'(\xi^-, \xi) = 1

. For x < \xi,

G(x, \xi) = A(\xi) \sin(x)

, so

G'(x, \xi) = A(\xi) \cos(x)

. For x > \xi,

G(x, \xi) = A(\xi) \sin(x)

, so

G'(x, \xi) = A(\xi) \cos(x)

step 13

x = \xi

G'(\xi^+, \xi) = A(\xi) \cos(\xi)

and

G'(\xi^-, \xi) = A(\xi) \cos(\xi)

. Thus,

A(\xi) \cos(\xi) - A(\xi) \cos(\xi) = 1 \implies A(\xi) = \frac{1}{\cos(\xi)}

step 14

The Green's function is

G(x, \xi) = \begin{cases} \frac{\sin(x)}{\cos(\xi)} &amp; \text{for } x &lt; \xi \\ \frac{\sin(x)}{\cos(\xi)} &amp; \text{for } x &gt; \xi \end{cases}

Answer

The Green's function for the boundary value problem

y'' + y = 0

with

y(0) = 0

and

y(\pi) = 0

G(x, \xi) = \begin{cases} \frac{\sin(x)}{\cos(\xi)} &amp; \text{for } x &lt; \xi \\ \frac{\sin(x)}{\cos(\xi)} &amp; \text{for } x &gt; \xi \end{cases}

Key Concept

Green's function for boundary value problems

Explanation

The Green's function is a fundamental solution used to solve inhomogeneous differential equations subject to boundary conditions. It represents the response of the system at a point

x

due to a unit impulse applied at another point

\xi