###### Elementary Differential Equations and Boundary Value Problems

195 SOLVED PROBLEMS

Question: 11.6.1

## Let f(x) = 1 for 0 < x < 1. Expand f(x) using the eigenfunctions (13), and discuss the pointwise and mean square convergence of the resulting series. ...

The series has the form (15), f(x)=\sum_{m=...
Question: 11.5.1

## Find the solution of the two-dimensional wave equation (3) in the unit circle with boundary condition (4) and initial conditions (5). ...

Assuming that u(r, t) = R(r)T(t), and substituting...
Question: 11.4.2

## The singular Sturm-Liouville boundary value problem −(xy′)′ = λxy, 0 < x < 1, y and y′ bounded as x → 0, y(1) = 0 has eigenfunctions 𝜙n(x) = J0(√λn x). (a) Show that the 𝜙n satisfy the orthogonality relation ∫0¹ x𝜙m(x)𝜙n(x)dx = 0, m ≠ n (21) with respect to the weight function r(x) = x. (b) ...

(a) From the differential equation, we see that p(...
Question: 11.4.1

## Show that the singular Sturm-Liouville boundary value problem consisting of the differential equation xy′′ + y′ + λxy = 0 with boundary conditions that both y and y′ remain bounded as x approaches 0 from the right and that β1y(1) + β2y′(1) = 0 is self-adjoint. ...

From the differential equation, we see that p(x) =...
Question: 11.3.2

## Find the solution of the heat conduction problem ut = uxx + xe^−t , 0 < x 0, (41) u(0, t) = 0, ux(1, t) + u(1, t) = 0, t > 0, (42) u(x, 0) = 0, 0 < x < 1. (43) ...

Again, we use the eigenvalues \lambda_{n}[/...
Question: 10.5.1

## Find the temperature u(x, t) at any time in a metal rod 50 cm long, insulated on the sides, which initially has a uniform temperature of 20° C throughout and whose ends are maintained at 0° C for all t > 0. ...

The temperature in the rod satisfies the heat cond...
Question: 10.4.2

## Suppose that f(x) = {1 − x, 0 < x ≤ 1, 0, 1 < x ≤ 2. (13) As indicated previously, we can represent f either by a cosine series or by a sine series. Sketch the graph of three periods of the sum of each of these series for −6 ≤ x ≤ 6. ...

In this example, L = 2, so the cosine series for f...
Question: 10.3.1

## Let f(x) = {0, −L < x < 0, L , 0 < x < L , (5) and let f be defined outside this interval so that f(x + 2L) = f(x) for all x. We will temporarily leave open the definition of f at the points x = 0, ±L. Find the Fourier series for this function and determine where it converges. ...

Three periods of the graph of y = f(x) are shown i...
Question: 10.2.3

## Consider again the function in Example 1 and its Fourier series (20). Investigate the speed with which the series converges. In particular, determine how many terms are needed so that the error is no greater than 0.01 for all x. ...

The $m^{th}$ partial sum in this seri...