Question

For what values of a
(if any) does the boundary value problem

x'' + ax = 0, x(0) = 0, x(π) = 0

have nontrivial (i.e. nonzero) solutions

Hint: In order to
solve, divide the problem into three cases

1. If a > 0. In this case let a = b^2. 2. If a < 0. In this
case let a = −b^2. 3. If a = 0.

Answer #1

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For what values of a (if any) does the boundary value
problem
x'' + ax' = 0, x(0) = 0, x(π) = 0
have nontrivial (i.e. nonzero) solutions
Hint: In order to solve, divide the problem into three
cases
1. If a > 0. In this case let a = b^2. 2. If a < 0. In this
case let a = −b^2. 3. If a = 0.

Solve the given boundary-value problem.
y'' − 2y' + 2y = 2x − 2,
y(0) = 0, y(π) = π

Find all eigenvalues and corresponding eigenfunctions for the
following boundary value problem (x^2)y'' + λy = 0, (1 < x <
2), y(1) = 0 = y(2) and in particular the three cases μ < 1/2, μ
= 1/2, and μ > 1/2 associated with the sign and vanishing of the
discriminant of the characteristic equation

Find u(x,y) harmonic in S with given boundary values: S =
{(x,y): 1 < y < 3} , u(x,y) = 5 (if y=1) and = 7 (when
y=3)
I have this problem to solve, and I'm not sure where to start.
Any help would be appreciated. Thanks!

Solve the given initial-value problem. (Enter the first three
nonzero terms of the solution.)
(x + 2)y'' +
3y = 0, y(0) =
0, y'(0) = 1

solve the boundary value problem:
y''(x)+y(x)=e^x for 0<x<pi with y(0)=0 and
y(pi)+y'(pi)=0.
please show all steps.

Given: The following boundary value problem:
y"+ lamda*y = 0;
0 < x < 2; y(0) =
0; y’(2) = 0
Find corresponding eigenvalues, (lamda)n
and normalized eigenfunctions yn
Expand the function f(x) = x, in terms of the eigen functions
obtained in (i)

1)Consider the following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1. Let af/ax = (x + y)2 =
x2 + 2xy + y2. Integrate each term of this
partial derivative with respect to x, letting
h(y) be an unknown function in y.
f(x, y) = + h(y)
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt + (8y + 6t
− 1) dy...

1) Consider the following initial-value problem.
(x + y)2 dx + (2xy + x2 − 2) dy =
0, y(1) = 1
Let af/ax = (x + y)2 = x2 + 2xy +
y2.
Integrate each term of this partial derivative with respect to
x, letting h(y)
be an unknown function in y.
f(x, y) = + h(y)
Find the derivative of h(y).
h′(y) =
Solve the given initial-value problem.
2) Solve the given initial-value problem.
(6y + 2t − 3)
dt...

Let a, c be positive constants and assume that a/ 2πc is a
positive integer. Consider the equation Utt +
aut = c^2Uxx , which represents a damped
version of the wave equation (telegrapher’s equation). Assuming
Dirichlet boundary conditions u(0, t) = u(1, t) = 0, on the
infinite strip 0 ≤ x ≤ 1, t ≥ 0, with initial conditions u(x, 0) =
f(x), ut(x, 0) = 0, complete the following:
(a) Find all separable solutions (of the form...

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