Solve the differential equation in Exercises 9–22.
15. √x (dy/dx) = e^(y+√x), x > 0

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Rewrite the given differential equation to isolate \( \frac{dy}{dx} \): \[ \sqrt{x} \frac{dy}{dx} = e^{y + \sqrt{x}} \implies \frac{dy}{dx} = \frac{e^{y + \sqrt{x}}}{\sqrt{x}} \].

Express the right-hand side as a product of exponentials to separate variables more easily: \[ \frac{dy}{dx} = \frac{e^y \cdot e^{\sqrt{x}}}{\sqrt{x}}. \]

Rewrite the equation in differential form to separate variables \( y \) and \( x \): \[ \frac{dy}{e^y} = \frac{e^{\sqrt{x}}}{\sqrt{x}} dx. \]

Integrate both sides: \[ \int \frac{1}{e^y} dy = \int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx. \] The left integral simplifies to \( \int e^{-y} dy \).

For the right integral, use substitution: let \( t = \sqrt{x} \), so \( x = t^2 \) and \( dx = 2t dt \). Substitute and simplify the integral accordingly before integrating.

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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 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 √x (dy/dx) = e^(y+√x).

Integration Techniques

Solving separable equations requires integrating functions of x and y separately. Familiarity with integrating exponential functions and substitutions, such as u = √x, helps simplify the integrals and find the general solution.

Implicit and Explicit Solutions

After integration, solutions may be implicit or explicit. Understanding how to manipulate and interpret these forms is important, especially when the solution involves expressions like e^(y+√x), which may require logarithmic manipulation to isolate y.

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