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 9.1.3

Chapter 9, Problem 9.1.3

Does the function y(t) = 2t satisfy the differential equation y'''(t) + y'(t) = 2?

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Identify the given function: \(y(t) = 2t\).

Compute the first derivative \(y'(t)\) by differentiating \(y(t)\) with respect to \(t\): \(y'(t) = \frac{d}{dt}(2t)\).

Compute the third derivative \(y'''(t)\) by differentiating \(y'(t)\) two more times: \(y'''(t) = \frac{d^2}{dt^2} y'(t)\).

Substitute \(y'(t)\) and \(y'''(t)\) into the differential equation \(y'''(t) + y'(t) = 2\) to check if the equation holds true.

Simplify the expression after substitution and verify whether the left-hand side equals the right-hand side for all \(t\).

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Key Concepts

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

Higher-Order Derivatives

Higher-order derivatives are derivatives of derivatives, such as the second derivative (rate of change of the rate of change) and the third derivative. In this problem, y'''(t) denotes the third derivative of y with respect to t, which must be computed to verify the differential equation.

Differential Equations

A differential equation relates a function and its derivatives. To check if a function satisfies a differential equation, substitute the function and its derivatives into the equation and verify if the equality holds for all values of the independent variable.

Derivative of a Linear Function

The derivative of a linear function y(t) = mt + b is a constant m, and higher derivatives of a linear function are zero. This property simplifies evaluating y'(t) and y'''(t) for the given function, aiding in checking the differential equation.

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Textbook Question

A family of exponential functions

b. Verify that the arc length of the curve y=f(x) on the interval [0, ln 2] is A(2^a-1) - 1/4a²A (2^-a - 1).

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Textbook Question

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form

a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t).

Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems.

(t² + 1)y′(t) + 2ty = 3t², y(2) = 8

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Textbook Question

23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.

B′(t) = 0.004B − 800, B(0) = 40,000

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Textbook Question

11–16. Initial value problems Solve the following initial value problems.

y'(x) = −y + 2, y(0) = −2

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Textbook Question

Stability of Euler's method Consider the initial value problem y′(t) = −ay, y(0) = 1 where a > 0; it has the exact solution y(t) = e⁻ᵃᵗ, which is a decreasing function.

a. Show that Euler's method applied to this problem with time step h can be written u₀ = 1, uₖ₊₁ = (1 − ah)uₖ for k = 0, 1, 2, ...

b. Show by substitution that uₖ = (1 − ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

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Textbook Question

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = yeᵗ, y(0) = −1

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