Solve the differential equation in Exercises 9–22.
12. (dy/dx) = 3x²e^(-y)

Verified step by step guidance

1. Rewrite the given differential equation \( \frac{dy}{dx} = 3x^{2} e^{-y} \) to separate the variables \( y \) and \( x \). Multiply both sides by \( e^{y} \) and multiply both sides by \( dx \) to get \( e^{y} dy = 3x^{2} dx \).
2. Integrate both sides separately: integrate \( e^{y} \) with respect to \( y \) and integrate \( 3x^{2} \) with respect to \( x \). This gives \( \int e^{y} dy = \int 3x^{2} dx \).
3. Compute the integrals: the integral of \( e^{y} \) with respect to \( y \) is \( e^{y} \), and the integral of \( 3x^{2} \) with respect to \( x \) is \( x^{3} + C \), where \( C \) is the constant of integration.
4. Write the implicit solution as \( e^{y} = x^{3} + C \).
5. If desired, solve explicitly for \( y \) by taking the natural logarithm of both sides: \( y = \ln(x^{3} + C) \), noting the domain restrictions for the logarithm.

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 expressions involving exponential functions, such as e^(-y), requires understanding the chain rule in reverse. Recognizing that the integral of e^(-y) with respect to y is -e^(-y) plus a constant is essential for solving the equation.

Implicit vs. Explicit Solutions

Solutions to differential equations can be implicit, where y is not isolated, or explicit, where y is expressed as a function of x. Understanding how to interpret and manipulate implicit solutions is important when the equation cannot be easily solved for y.