Chapter 1: Infinite Series, Power Series

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.

7.${\mathbf{\int }}_{{\mathbf{x}}_{\mathbf{1}}}^{{\mathbf{x}}_{\mathbf{2}}}{\mathit{e}}^{\mathbf{x}}\sqrt{\mathbf{1}\mathbf{+}\mathbf{y}{\mathbf{\text{'}}}^{\mathbf{2}}}\mathit{d}\mathit{x}$Hint: In the last integration, let$\mathit{u}\mathbf{=}{\mathit{e}}^{\mathbf{x}}$ and see Chapter 5, Problem 1.6.

Prove that a particle constrained to stay on a surface $\mathbf{f}\left(x,y,z\right)\mathbf{=}\mathbf{0}$, but subject to no other forces, moves along a geodesic of the surface. Hint: The potential energy$\mathbf{V}$is constant, since constraint forces are normal to the surface and so dono work on the particle. Use Hamilton’s principle and show that the problem offinding a geodesic and the problem of finding the path of the particle are identicalmathematics problems.

Do Problem 23 for a spherical cavity containing a constant source of heat. Use the same radii and temperatures as in Problem 23.

(Problem 23: Heat is escaping at a constant rate [$\frac{dQ}{dt}$in (1.1) is constant] through the walls of a long cylindrical pipe. Find the temperature T at a distance rfrom the axis of the cylinder if the inside wall has radius r=1 and temperature T=100 and the outside wall has r=2 and T=0)

Find the orthogonal trajectories of each of the following families of curves. In each case, sketch or computer plot several of the given curves and several of their orthogonal trajectories. Be careful to eliminate the constant fromfor the original curves; this constant takes different values for different curves ${\mathit{y}}^{\mathbf{\text{'}}}$of the original family, and you want an expression for${\mathit{y}}^{\mathbf{\text{'}}}$which is valid for all curves of the family crossed by the orthogonal trajectory you are trying to find. See equations (2.10)to (2.12).

$\mathit{x}\mathit{y}\mathbf{=}\mathit{k}$

Using (3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after()3.9, and Example 1.

14.$\frac{\text{dy}}{\text{dx}}=\frac{\text{3y}}{{\text{3y}}^{\text{2/3}}-\text{x}}$

Hint: For Problems 12to 14, solve forx in terms of y.

Find the general solution of(1.2)for an RLcircuit$\left(\frac{\text{1}}{\text{C}}=\text{0}\right)$with$\mathit{V}\mathbf{=}{\mathit{V}}_{\mathbf{\circ }}\mathit{c}\mathit{o}\mathit{s}\mathit{\omega }\mathit{t}\left(\omega =constant\right)$.

Do Problems 16and 17using$\mathit{V}\mathbf{=}{\mathit{V}}_{\mathbf{0}}{\mathit{e}}^{\mathbf{i}\mathbf{w}\mathbf{t}}$, and find the solutions for16 and17 by taking real parts of the complex solutions.

Extend the radioactive decay problem (Example 2) one more stage, that is, let ${\mathit{\lambda }}_{\mathbf{3}}$be the decay constant of polonium and find how much polonium there is at time t.

Using(3.9), find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after(3.9), and Example 1

$\mathit{d}\mathit{y}\mathbf{+}\left(2xy-x{e}^{-x^2}\right)\mathit{d}\mathit{x}\mathbf{=}\mathbf{0}$

Test for convergence: $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{2}}^{\mathbf{\infty }}\frac{\mathbf{2}{\mathbf{n}}^{\mathbf{3}}}{{\mathbf{n}}^{\mathbf{4}}\mathbf{-}\mathbf{2}}$