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.3

Chapter 9, Problem 9.PE.3

In Exercises 1–22, solve the differential equation.

sec x dy + x cos² y dx = 0

Verified step by step guidance

Rewrite the given differential equation in the form \(M(x,y)\,dx + N(x,y)\,dy = 0\). Here, the equation is \(\sec x \, dy + x \cos^{2} y \, dx = 0\), so rearranged it becomes \(x \cos^{2} y \, dx + \sec x \, dy = 0\).

Check if the differential equation is exact by computing the partial derivatives \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\), where \(M = x \cos^{2} y\) and \(N = \sec x\).

If the equation is not exact, look for an integrating factor that depends on either \(x\) or \(y\) to make it exact. This often involves checking if \(\frac{1}{N} (\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})\) is a function of only one variable.

Once the equation is exact (or made exact), find the potential function \(\Psi(x,y)\) such that \(\frac{\partial \Psi}{\partial x} = M\) and \(\frac{\partial \Psi}{\partial y} = N\) by integrating \(M\) with respect to \(x\) and then determining the function of \(y\).

Write the implicit solution as \(\Psi(x,y) = C\), where \(C\) is a constant, representing the general solution to the differential equation.

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Key Concepts

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

Differential Equations

A differential equation relates a function with its derivatives. Solving it means finding the function that satisfies this relationship. In this problem, the equation involves derivatives of y with respect to x, and the goal is to find y as a function of x.

Separable Equations

A separable differential equation can be written so that all terms involving y are on one side and all terms involving x are on the other. This allows integration of each side independently. Recognizing separability is key to solving equations like the one given.

Integration Techniques

Solving separable equations requires integrating functions of x and y separately. Familiarity with integrating trigonometric functions such as sec x and cos² y is essential. Proper integration leads to an implicit or explicit solution of the differential equation.

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Related Practice

Textbook Question

In Exercises 1–22, solve the differential equation.

y' = xeʸ√(x-2)

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Textbook Question

In Exercises 23–28, solve the initial value problem.

y dx + (3x - xy + 2)dy = 0, y(2) = -1, y < 0

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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.

b. How many pounds of salt are in the tank after 1 minute? after 30 minutes?

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Textbook Question

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x²+y²)dx + xy dy = 0

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Textbook Question

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x.exp(y/x) + y)dx - x dy = 0

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Textbook Question

In Exercises 1–22, solve the differential equation.