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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 9.1.15
Chapter 9, Problem 9.1.15

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.
u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³

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Identify the given function and the differential equation: the function is \(u(t) = C_1 t^5 + C_2 t^{-4} - t^3\), and the differential equation is \(t^2 u''(t) - 20 u(t) = 14 t^3\).
Compute the first derivative \(u'(t)\) by differentiating each term of \(u(t)\) with respect to \(t\): use the power rule for derivatives.
Compute the second derivative \(u''(t)\) by differentiating \(u'(t)\) again with respect to \(t\).
Substitute \(u(t)\) and \(u''(t)\) into the left-hand side of the differential equation: calculate \(t^2 u''(t) - 20 u(t)\).
Simplify the expression obtained after substitution and verify if it equals the right-hand side \$14 t^3\( for all \)t\(, confirming that \)u(t)$ is a solution.

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Key Concepts

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

General Solution of Differential Equations

A general solution includes arbitrary constants representing all possible solutions to a differential equation. Verifying a solution involves substituting it into the equation to check if it satisfies the relationship for all values in the domain.
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Solutions to Basic Differential Equations

Second Derivative and Differentiation

The second derivative measures the rate of change of the first derivative and is essential in second-order differential equations. Calculating u''(t) accurately is crucial for substituting into the given differential equation.
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The Second Derivative Test: Finding Local Extrema

Substitution and Simplification in Differential Equations

To verify a solution, substitute the function and its derivatives into the differential equation. Simplify the resulting expression to confirm if both sides of the equation are equal, ensuring the function satisfies the equation.
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Related Practice
Textbook Question

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

p'(x) = 16/x⁹ - 5 + 14x⁶

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Textbook Question

15–16. {Use of Tech} Solving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P₀ the initial population.


r=0.2, K=300, P₀=50

Textbook Question

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.


Textbook Question

33–42. Solving initial value problems Solve the following initial value problems.

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

Textbook Question

Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.

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Textbook Question

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

(t² + 1)³yy'(t) = t(y² + 4)