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 9.3.36

Chapter 9, Problem 9.3.36

33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.
yy'(x) = 2x/(2 + y)², y(1) = −1

Verified step by step guidance

Rewrite the given differential equation in terms of \( y \) and \( x \). The equation is \( y \frac{dy}{dx} = \frac{2x}{(2 + y)^2} \). Our goal is to separate variables if possible.

Express \( \frac{dy}{dx} \) explicitly: \( \frac{dy}{dx} = \frac{2x}{y(2 + y)^2} \). This suggests separating variables by bringing all \( y \)-terms to one side and \( x \)-terms to the other.

Rewrite the equation as \( y(2 + y)^2 dy = 2x dx \). This sets up the integral form where we integrate both sides with respect to their variables.

Integrate both sides: \( \int y(2 + y)^2 dy = \int 2x dx \). You will need to expand \( (2 + y)^2 \) and then multiply by \( y \) before integrating the left side.

After integrating, apply the initial condition \( y(1) = -1 \) to solve for the constant of integration. The resulting equation will be an implicit solution involving \( x \) and \( y \).

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Key Concepts

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

Implicit Differentiation and Implicit Solutions

Implicit differentiation involves finding derivatives when the function is not explicitly solved for one variable. In this problem, the solution is left in implicit form, meaning y is defined by an equation involving both x and y, rather than y expressed explicitly as a function of x.

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition, y(1) = -1, helps identify the unique solution curve among potentially many implicit solutions.

Use of Technology for Graphing Implicit Solutions

Graphing software can plot implicit solutions that are difficult to solve explicitly. It helps visualize the solution curve and distinguish which branch corresponds to the initial condition, especially when the implicit equation defines multiple functions.

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Finding The Implicit Derivative Example 1

Related Practice

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.

y'(t) = eʸᐟ²sin t

Textbook Question

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem

P'(t) = rP (1-P/K), P(0) = P₀

is P(t) = K/((K/P₀ − 1)e⁻ʳᵗ + 1)

Textbook Question

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

u''(x) = 55x⁹ + 36x⁷ - 21x⁵ + 10x⁻³

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Textbook Question

20–22. {Use of Tech} Solving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M₀. Then graph the solution to be sure that M(0) and lim(t→∞) M(t) are correct.

r = 0.05, K = 1200, M₀ = 90

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Textbook Question

Consider the differential equation y'(t) = t² - 3y² and the solution curve that passes through the point (3, 1). What is the slope of the curve at (3, 1)?

Textbook Question

What is the equilibrium solution of the equation y'(t) = 3y − 9? Is it stable or unstable?

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