Solve the differential equation in Exercises 9–22.
11. (dy/dx) = e^(x-y)

Verified step by step guidance

1. Rewrite the given differential equation \( \frac{dy}{dx} = e^{x - y} \) to separate the variables \( y \) and \( x \). Start by expressing \( e^{x - y} \) as \( e^x \cdot e^{-y} \), so the equation becomes \( \frac{dy}{dx} = e^x e^{-y} \).
2. Separate the variables by multiplying both sides by \( e^y \) and multiplying both sides by \( dx \), giving \( e^y dy = e^x dx \). This puts all \( y \)-terms on one side and all \( x \)-terms on the other.
3. Integrate both sides: integrate \( e^y \) with respect to \( y \) on the left, and integrate \( e^x \) with respect to \( x \) on the right. This yields \( \int e^y dy = \int e^x dx \).
4. After integrating, write the general solution including the constant of integration \( C \). The integrals of \( e^y \) and \( e^x \) are both \( e^y \) and \( e^x \) respectively, so the equation becomes \( e^y = e^x + C \).
5. If needed, solve explicitly for \( y \) by taking the natural logarithm of both sides, resulting in \( y = \ln(e^x + C) \), which represents the implicit or explicit general solution to the differential equation.

Key Concepts

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 of Exponential Functions

Integrating exponential functions involves recognizing the form e^(u) and applying substitution if necessary. The integral of e^(u) with respect to u is e^(u) plus a constant, which is essential when solving differential equations involving exponentials.

Implicit and Explicit Solutions

Solutions to differential equations can be implicit, where y is defined implicitly by an equation involving both x and y, or explicit, where y is expressed directly as a function of x. Understanding both forms helps interpret and verify solutions.