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.1

Chapter 9, Problem 9.PE.1

In Exercises 1–22, solve the differential equation.
y' = xeʸ√(x-2)

Verified step by step guidance

Rewrite the given differential equation as \(\frac{dy}{dx} = x e^{y} \sqrt{x - 2}\) to clearly identify the variables and their derivatives.

Separate the variables by dividing both sides by \(e^{y}\) and multiplying both sides by \(dx\), giving \(e^{-y} dy = x \sqrt{x - 2} \, dx\).

Integrate both sides separately: integrate \(e^{-y} dy\) with respect to \(y\) on the left side, and integrate \(x \sqrt{x - 2} \, dx\) with respect to \(x\) on the right side.

For the right side integral, consider using a substitution such as \(u = x - 2\) to simplify the integral \(\int x \sqrt{x - 2} \, dx\).

After integrating both sides, include the constant of integration \(C\) and solve for \(y\) if possible to express the general solution.

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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 Techniques

Solving separable equations requires integrating both sides after separation. Familiarity with integration methods, including substitution and handling functions like exponential and square roots, is essential to evaluate the integrals correctly.

Domain Considerations for the Solution

The presence of √(x-2) restricts the domain to x ≥ 2 to keep the expression real. Understanding domain constraints ensures the solution is valid and helps identify any initial conditions or intervals where the solution applies.

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Related Practice

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.

y' = y/x + cos ((y-x)/x)

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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.