In Exercises 1–22, solve the differential equation.

y' = xeˣ⁻ʸ csc y

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Rewrite the given differential equation \(y' = xe^{x - y} \csc y\) as \(\frac{dy}{dx} = xe^{x - y} \csc y\) to clearly identify the derivative form.

Express the right side to separate variables involving \(y\) and \(x\). Notice that \(e^{x - y} = e^x \cdot e^{-y}\), so rewrite the equation as \(\frac{dy}{dx} = x e^x e^{-y} \csc y\).

Rewrite \(\csc y\) as \(\frac{1}{\sin y}\) and combine it with \(e^{-y}\) to get \(e^{-y} \csc y = \frac{e^{-y}}{\sin y}\). Then, rearrange the equation to isolate \(dy\) and \(dx\) terms: \(\sin y e^y dy = x e^x dx\).

Separate variables by multiplying both sides appropriately to get \(\sin y e^y dy = x e^x dx\), which allows integration on both sides with respect to their own variables.

Integrate both sides: compute \(\int \sin y e^y dy\) on the left and \(\int x e^x dx\) on the right. After integration, include the constant of integration and express the implicit 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 functions of x and y separately. Familiarity with integrating exponential functions, trigonometric functions like cosecant, and their combinations is essential to find the implicit or explicit solution.

Implicit vs. Explicit Solutions

After integration, solutions may be implicit (involving both x and y) or explicit (solved for y). Understanding how to interpret and manipulate implicit solutions is important, especially when the equation involves transcendental functions like 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.

y' = sin³ x cos² y

Textbook Question

In Exercises 1–22, solve the differential equation.

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

Textbook Question