Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.1.50d

Chapter 9, Problem 9.1.50d

A second-order equation Consider the differential equation y''(t) - k²y(t) = 0 where k > 0 is a real number.

d. For a positive real number k, verify that the general solution of the equation may also be expressed in the form y(t) = C₁cosh(kt) + C₂sinh(kt), where cosh and sinh are the hyperbolic cosine and hyperbolic sine, respectively (Section 7.3).

Verified step by step guidance

Start with the given differential equation: \(y''(t) - k^{2} y(t) = 0\), where \(k > 0\).

Write the characteristic equation associated with the differential equation: \(r^{2} - k^{2} = 0\).

Solve the characteristic equation for \(r\): \(r^{2} = k^{2}\), so \(r = \pm k\).

Write the general solution using the exponential functions corresponding to the roots: \(y(t) = C_{1} e^{kt} + C_{2} e^{-kt}\).

Recall the definitions of hyperbolic cosine and sine: \(\cosh(kt) = \frac{e^{kt} + e^{-kt}}{2}\) and \(\sinh(kt) = \frac{e^{kt} - e^{-kt}}{2}\). Use these to rewrite the general solution as \(y(t) = A \cosh(kt) + B \sinh(kt)\), where \(A\) and \(B\) are constants related to \(C_{1}\) and \(C_{2}\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Second-Order Linear Differential Equations

These are differential equations involving the second derivative of a function and can often be solved by finding characteristic equations. Solutions typically involve exponential, trigonometric, or hyperbolic functions depending on the nature of the roots of the characteristic equation.

Characteristic Equation and Its Roots

For the equation y'' - k²y = 0, the characteristic equation is r² - k² = 0, yielding roots r = ±k. These roots determine the form of the general solution, which can be expressed using exponentials or hyperbolic functions when roots are real and distinct.

Hyperbolic Functions (cosh and sinh)

Hyperbolic cosine and sine, defined as cosh(t) = (e^t + e^{-t})/2 and sinh(t) = (e^t - e^{-t})/2, provide an alternative way to express solutions of differential equations with real roots. They are especially useful for rewriting solutions involving exponentials in a more compact and interpretable form.

Recommended video:

5:50

Asymptotes of Hyperbolas

Related Practice

Textbook Question

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

Textbook Question

A physiological model A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time t = 0, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant.

d. After how many minutes does the drug mass reach 90% of its steady-state level?

Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

d. Compare the errors in the approximations to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

views

Textbook Question

17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

c. Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing?

y'(t) = cos y for |y| ≤ π

views

Textbook Question

[Use of Tech] Analysis of a separable equation Consider the differential equation yy'(t) = ½eᵗ + t and carry out the following analysis.

c. Graph the solutions in part (b) and describe their behavior as t increases.

Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

d. In general, how does halving the time step affect the error at t=0.2 and t=0.4?

y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²