Ch. 9 - First-Order Differential Equations

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 9 - First-Order Differential Equations

Problem 9.PE.17

Chapter 9, Problem 9.PE.17

In Exercises 1–22, solve the differential equation.

y' = sin³ x cos² y

Verified step by step guidance

Recognize that the given differential equation is separable since it can be written as a product of a function of x and a function of y: \(y' = \sin^{3} x \cos^{2} y\).

Rewrite the differential equation in differential form: \(\frac{dy}{dx} = \sin^{3} x \cos^{2} y\).

Separate the variables by dividing both sides by \(\cos^{2} y\) and multiplying both sides by \(dx\): \(\frac{dy}{\cos^{2} y} = \sin^{3} x \, dx\).

Express the left side in terms of \(\sec^{2} y\) to facilitate integration: \(\sec^{2} y \, dy = \sin^{3} x \, dx\).

Integrate both sides separately: \(\int \sec^{2} y \, dy = \int \sin^{3} x \, dx\), then solve each integral using appropriate techniques (such as substitution or trigonometric identities).

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Key Concepts

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

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 solution.

Integration of Trigonometric Functions

Solving the equation involves integrating powers of sine and cosine functions. Familiarity with trigonometric identities and integration techniques, such as substitution or power-reduction formulas, is essential to evaluate these integrals correctly.

Implicit Solutions and Initial Conditions

After integration, solutions may be implicit, involving both x and y in an equation. Understanding how to interpret implicit solutions and apply initial conditions, if given, helps in finding explicit solutions or particular solutions to the differential equation.

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Initial Value Problems

Related Practice

Textbook Question

28. Derivation of Equation (7) in Example 4

a. Show that the solution of the equation

di /dt + R/Li = V/L

i = V/R + Cexp(-(R/L)i) .

Textbook Question

In Exercises 1–22, solve the differential equation.

y' = xeˣ⁻ʸ csc y

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Textbook Question

In Exercises 43 and 44, let S represent the pounds of salt in a tank at time t minutes. Set up a differential equation representing the given information and the rate at which S changes. Then solve for S and answer the particular questions.

Pure water flows into a tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the rate of 5 gal/min. The tank initially holds 200 gal of solution containing 50 pounds of salt.

c. When will the tank have exactly 5 pounds of salt and how many gallons of solution will be in the tank?

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Textbook Question

In Exercises 1–22, solve the differential equation.

(x + 3y²) dy + y dx = 0 (Hint: d(xy) = y dx + x dy)

Textbook Question

In Exercises 1–22, solve the differential equation.

xy' + 2y = 1 - x⁻¹

Textbook Question

Solve the following initial value problem for u as a function of t:

du/dt + (k/m) u = 0 (k and m positive constants), u(0) = u₀

a. as a first-order linear equation.