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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.1.9
Chapter 9, Problem 9.1.9

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.
y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0

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Identify the given function and the differential equation: the function is \(y(t) = C_1 \sin 4t + C_2 \cos 4t\), and the differential equation is \(y''(t) + 16y(t) = 0\).
Compute the first derivative \(y'(t)\) of the function using the chain rule: differentiate each term with respect to \(t\).
Compute the second derivative \(y''(t)\) by differentiating \(y'(t)\) again with respect to \(t\).
Substitute \(y(t)\) and \(y''(t)\) into the differential equation \(y''(t) + 16y(t)\) and simplify the expression.
Check if the simplified expression equals zero for all \(t\), which verifies that the given function is indeed a solution to the differential equation.

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Key Concepts

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

General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the complete set of functions that satisfy the equation, allowing for initial conditions to specify a unique solution.
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Solutions to Basic Differential Equations

Verification by Substitution

To verify a solution, substitute the given function and its derivatives into the differential equation. If the equation holds true for all values in the domain, the function is a valid solution.
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Substitution With an Extra Variable

Second Derivative and Trigonometric Functions

Calculating the second derivative of trigonometric functions like sine and cosine involves applying differentiation rules twice. For example, the second derivative of sin(kt) is -k² sin(kt), which is essential in solving and verifying solutions of differential equations involving harmonic motion.
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Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question

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

y'(t) = y³sin t, y(0) = 1

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Textbook Question

17–20. Verifying solutions of initial value problems Verify that the given function y is a solution of the initial value problem that follows it.

y(t) = 8t⁶ - 3; ty'(t) - 6y(t) = 18, y(1) = 5

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Textbook Question

5–10. First-order linear equations Find the general solution of the following equations.


v'(y) − v/2 = 14

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Textbook Question

Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²

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Textbook Question

45–48. General first-order linear equations Consider the general first-order linear equation y'(t)+a(t)y(t)=f(t). This equation can be solved, in principle, by defining the integrating factor p(t)=exp(∫a(t)dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes


p(t)(y′(t) + a(t)y(t)) = d/dt(p(t)y(t)) = p(t)f(t).


Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.


y′(t) + (2t)/(t² + 1)y(t) = 1 + 3t², y(1) = 4

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Textbook Question

21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.

y'(t) = t lnt + 1

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