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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 9 - Differential EquationsProblema 9.R.16
Capitolo 9, Problema 9.R.16

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(x) = 4x csc y, y(0) = π/2

Guida verificata passo dopo passo
1
Identify the type of differential equation given: \( y'(x) = 4x \csc y \). Notice that it is separable because the right-hand side can be expressed as a product of a function of \( x \) and a function of \( y \).
Rewrite the differential equation in separable form by expressing \( y' = \frac{dy}{dx} \) and rearranging terms to isolate \( y \) and \( x \) on opposite sides: \( \sin y \, dy = 4x \, dx \).
Integrate both sides: compute \( \int \sin y \, dy \) on the left and \( \int 4x \, dx \) on the right. Remember to include the constant of integration after integrating.
Apply the initial condition \( y(0) = \frac{\pi}{2} \) to solve for the constant of integration. Substitute \( x = 0 \) and \( y = \frac{\pi}{2} \) into the integrated equation.
Rewrite the implicit solution to express \( y \) explicitly in terms of \( x \) if possible, or leave it in implicit form as the solution to the initial value problem.

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Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.
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Solving Separable Differential Equations

Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition helps determine the unique solution by solving for the constant of integration after finding the general solution.
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Initial Value Problems

Integration of Trigonometric Functions

Solving the given differential equation involves integrating expressions containing trigonometric functions like cosecant (csc y). Understanding how to integrate these functions, often using identities or substitution, is essential to find the explicit solution.
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Introduction to Trigonometric Functions
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Domanda del libro di testo

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.

c. When does the temperature of the coffee reach 50°C?

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Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.

c. After how many hours does the population reach half of the carrying capacity

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A second-order equation Consider the equation


t² y′′(t) + 2ty′(t) − 12y(t) = 0


b. Assuming the general solution of the equation is


y(t) = C₁ tᵖ¹ + C₂ tᵖ²,


find the solution that satisfies the conditions


y(1) = 0, y′(1) = 7

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Domanda del libro di testo

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.

y′(x) = x/y, y(2) = 4

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Domanda del libro di testo

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = (2t+1)(y²+1)

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Domanda del libro di testo

A first-order equation Consider the equation t² y′(t) + 2ty(t) = e⁻ᵗ

a. Show that the left side of the equation can be written as the derivative of a single term.

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