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.15

Chapter 9, Problem 9.PE.15

In Exercises 1–22, solve the differential equation.

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

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Rewrite the given differential equation \((x + 3y^{2}) \, dy + y \, dx = 0\) to isolate terms involving \(dx\) and \(dy\). This can be expressed as \(y \, dx + (x + 3y^{2}) \, dy = 0\).

Recognize the hint \(d(xy) = y \, dx + x \, dy\) and try to express part of the equation in terms of \(d(xy)\) to simplify the problem.

Rewrite the equation as \(y \, dx + x \, dy + 3y^{2} \, dy = 0\), which becomes \(d(xy) + 3y^{2} \, dy = 0\).

Integrate both sides with respect to the appropriate variables: integrate \(d(xy)\) directly and integrate \(3y^{2} \, dy\) separately.

After integration, combine the results to form an implicit solution involving \(xy\) and \(y^{3}\), then solve for the general solution of the differential equation.

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Key Concepts

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

Exact Differential Equations

An exact differential equation can be written in the form M(x,y) dx + N(x,y) dy = 0, where there exists a function F(x,y) such that dF = M dx + N dy. Solving involves finding F(x,y) = C, which implicitly defines the solution. Recognizing exactness often simplifies solving differential equations.

Using the Product Rule for Differentials

The product rule states d(uv) = u dv + v du for functions u and v. In this problem, the hint d(xy) = y dx + x dy helps rewrite terms to identify exact differentials or simplify the equation, facilitating integration and solution.

Separation and Rearrangement of Variables

Rearranging terms to isolate dy and dx or grouping expressions helps in identifying integrable forms or exact differentials. This step is crucial for transforming the given equation into a solvable form, either by direct integration or applying known solution methods.

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Separation of Variables

Related Practice

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.

y' = sin³ x cos² y

Textbook Question

In Exercises 1–22, solve the differential equation.

xy' + 2y = 1 - x⁻¹

Textbook Question

In Exercises 1–22, solve the differential equation.

2y' - y = xe^(x/2)

Textbook Question