In Exercises 1–22, solve the differential equation.

y' = eʸ/xy

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Rewrite the given differential equation \(y' = \frac{e^y}{x y}\) in Leibniz notation as \(\frac{dy}{dx} = \frac{e^y}{x y}\) to clearly see the variables involved.

Separate the variables by multiplying both sides by \(y \, dy\) and \(x \, dx\) appropriately to isolate \(y\) terms on one side and \(x\) terms on the other side. This gives \(y \, e^{-y} \, dy = \frac{1}{x} \, dx\).

Integrate both sides: compute \(\int y e^{-y} \, dy\) on the left and \(\int \frac{1}{x} \, dx\) on the right. Use integration by parts for the left integral since it involves a product of \(y\) and \(e^{-y}\).

After integrating, include the constant of integration \(C\) on one side to represent the general solution of the differential equation.

Finally, express the implicit solution relating \(x\) and \(y\), or solve explicitly for \(y\) if possible, depending on the form of the integrated expression.

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주요 개념

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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 form 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 integrating exponential functions and rational expressions is essential to correctly evaluate the integrals and obtain 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 isolating y is difficult or impossible.

In Exercises 1–22, solve the differential equation.y' = eʸ/xy