Skip to main content
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 9.1.24
Chapter 9, Problem 9.1.24

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

Verified step by step guidance
1
Identify the given differential equation: \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^{6}\). This means the derivative of \(p(x)\) with respect to \(x\) is given by this expression.
To find the general solution \(p(x)\), integrate both sides of the equation with respect to \(x\): \(p(x) = \int \left( \frac{16}{x^{9}} - 5 + 14x^{6} \right) \, dx\).
Rewrite the integrand to make integration easier: \(\frac{16}{x^{9}} = 16x^{-9}\), so the integral becomes \(\int \left( 16x^{-9} - 5 + 14x^{6} \right) \, dx\).
Integrate each term separately using the power rule for integration: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
After integrating each term, combine the results and add the arbitrary constant of integration \(C\) to express the general solution \(p(x)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 contains arbitrary constants representing the family of curves satisfying the equation. For first-order equations, integrating the derivative function yields the general solution plus a constant of integration.
Recommended video:
04:00
Solutions to Basic Differential Equations

Integration of Functions

Integration is the reverse process of differentiation and is used to find the original function from its derivative. To solve p'(x), integrate each term separately with respect to x, applying power rule integration and adding an arbitrary constant.
Recommended video:
05:11
Integrals of General Exponential Functions

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like 16/x⁹ and 14x⁶ by rewriting negative exponents and applying the formula correctly.
Recommended video:
04:04
Power Rule for Indefinite Integrals
Related Practice
Textbook Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.


A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C 

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

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)

Textbook Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.


An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C? 

1
views
Textbook Question

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