Differential Equations and Boundary Value Problems: Computing and Modeling 5th Edition solutions

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Suppose that a crossbow bolt is shot straight upward with initial velocity 288 ft=s. If its deceleration due to air resistance is .0:04/v, then its height x.t / satisfies the initial value problem x00 D 32 .0:04/x0 I x.0/ D 0; x0 .0/ D 288: Find the maximum height that the bolt attains and the time r

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Three 100-gal fermentation vats are connected as indicated in Fig. 4.1.13, and the mixture in each tank is kept uniform by stirring. Denote by xi.t / the amount (in pounds) of alcohol in tank Ti at time t (i D 1, 2, 3). Suppose that the mixture circulates between the tanks at the rate of 10 gal=min.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
The incoming water has pollutant concentration \(\mathrm{c}(\mathrm{t})=10(1+\cos t)\mathrm{\ L}/\mathrm{m}^3\) that varies between 0 and 20, with an average concentration of 10 \(\mathrm{L} / \mathrm{m}^{3}\) and a period of oscillation of slightly over \(6 \frac{1}{4}\) months. Does it seem predict

Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 20, find a particular solution yp of the given equation. In all these problems, primes denote derivatives with respect to x.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
A building consists of two floors. The first floor is attached rigidly to the ground, and the second floor is of mass m D 1000 slugs (fps units) and weighs 16 tons (32,000 lb). The elastic frame of the building behaves as a spring that resists horizontal displacements of the second floor; it requires

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the linear systems in Problems 1 through 20. If initial conditions are given, find the particular solution that satisfies them. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given syst

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Express J4.x/ in terms of J0.x/ and J1.x/.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find general solutions of the systems in Problems 1 through 22. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Rework both parts of Problem 7, with the sole difference that the deceleration due to air resistance now is .0:001/v2 ft=s2 when the cars velocity is v feet per second.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Find the general solutions of the differential equations in Problems 1 through 20.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
A river 100 ft wide is flowing north at w feet per second. A dog starts at .100; 0/ and swims at v0 D 4 ft=s, always heading toward a tree at .0; 0/ on the west bank directly across from the dogs starting point. (a) If w D 2 ft=s, show that the dog reaches the tree. (b) If w D 4 ft=s, show that the d

Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 1 through 16, transform the given differential equation or system into an equivalent system of first-order differential equations

Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 17 through 34, use the method of variation of parameters (and perhaps a computer algebra system) to solve the initial value problem x0 D Ax C f.t /; x.a/ D xa: In each problem we provide the matrix exponential eAt as provided by a computer algebra system.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Solve the boundary value problems in Problems 1 through 12.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Consider the crossbow bolt of Example 3, shot straight upward from the ground (y D 0) at time t D 0 with initial velocity v0 D 49 m=s. Take g D 9:8 m=s2 and D 0:0011 in Eq. (12). Then use Eqs. (13) and (14) to show that the bolt reaches its maximum height of about 108:47 m in about 4:61 s.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
Use Laplace transforms to solve the initial value problems in Problems 1 through 16.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
A mass m hangs on the end of a cord around a pulley of radius a and moment of inertia I , as shown in Fig. 3.6.11. The rim of the pulley is attached to a spring (with constant k). Assume small oscillations so that the spring remains essentially horizontal and neglect friction. Find the natural circul

Differential Equations and Boundary Value Problems: Computing and Modeling 5
According to a newspaper account, a paratrooper survived a training jump from 1200 ft when his parachute failed to open but provided some resistance by flapping unopened in the wind. Allegedly he hit the ground at 100 mi=h after falling for 8 s. Test the accuracy of this account. (Suggestion: Find i

Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 11 through 26, the values of a period 2function f .t / in one full period are given. Sketch several periods of its graph and find its Fourier series.

Differential Equations and Boundary Value Problems: Computing and Modeling 5
In Problems 21 through 30, set up the appropriate form of a particular solution yp, but do not determine the values of the coefficients