Differential Equations with Linear Algebra by Matthew R. Boelkins, Jack L. Goldberg, Merle C. Potter | 9780195385861 | 0195385861 | Holooly

Differential Equations with Linear Algebra [EXP-50954]

156 SOLVED PROBLEMS

Question: 2.4.3

A can of soda at room temperature 70◦F is placed in a refrigerator that maintains a constant temperature of 40◦F. After 1 hour in the refrigerator, the temperature of the soda is 58◦F. At what time will the soda’s temperature be 41◦F?

Verified Answer:

Let T(t ) denote the temperature of the soda at ti...

Question: 2.7.1

A population P(t ) exhibits logistic growth according to the model (a) Determine the values of P for which P is an increasing function (b) Plot the direction field for the differential equation (c) Determine the value(s) of P for which P is increasing most rapidly (d) Solve the IVP explicitly for P

Verified Answer:

(a) To determine where P is increasing, we require...

Question: 2.4.2

A radioactive isotope initially has 40 g of mass. After 10 days of radioactive decay, its mass is 39.7 g. What is the isotope’s half-life? At what time t will 1 g remain?

Verified Answer:

Because the isotope decays radioactively, we know ...

Question: 3.4.3

and consider the system of differential equations given by x´=Ax. Find the general solution of the system, determine all equilibrium solutions to the system, and plot the direction field for the system. Include sketches of several trajectories and discuss the long-term behavior

Verified Answer:

We find that A has eigenvalues

λ_{1}[/latex...

Question: 8.3.1

By assuming that y has a power series expansion of the form y(t ) = a0 +a1t +a2t^2 +a3t^3+· · · , determine the solution to the initial-value problem y′= y, y(0) = 1

Verified Answer:

Writing

y(t ) = a_{0} +a_{1}t +a_{2}t^{2} +...

Question: 1.9.1

Compute the determinant of the matrix

Verified Answer:

By definition,

det \begin{bmatrix} 2 & ...

Question: 1.9.2

Compute the determinant of the matrix

Verified Answer:

Again using the definition, we see that

det...

Question: 5.2.2

Compute the Laplace transform of f (t ) = 1.

Verified Answer:

From the definition, we observe that

L[1] =...

Question: 5.2.1

Compute the Laplace transform of f (t ) = t .

Verified Answer:

By definition,

L[t]=\int_{0}^{∞}{te^{−st} d...

Question: 5.5.4

Consider a mass of 1 kg attached to a spring with spring constant k = 13 such that the system has damping constant c = 4. Assume that the mass is displaced 1 m from equilibrium and released at t = 0; furthermore, at time t =π the forcing function f (t )=2sin3t is applied. Assuming consistent units,

Verified Answer:

From our work with spring-mass systems, we know th...

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