Question

Find the general solution to the differential equation.\(y^{\prime}=e^{y} x^{2}\)

Short answer

The general solution is \( y(x) = -\ln(-\frac{x^3}{3} - C) \).

Step-by-step solution

Separate Variables

Start by rearranging the equation to separate the variables. We have the original differential equation: \( y' = e^y x^2 \). This can be rewritten as \( \frac{dy}{dx} = e^y x^2 \). To separate variables, rewrite it as \( e^{-y} dy = x^2 dx \). Now, both sides are in terms of a single variable.

Integrate Both Sides

Integrate both sides of the separated equation. For the left-hand side, integrate \( \int e^{-y} \, dy \). The integration gives \( -e^{-y} + C_1 \). For the right-hand side, integrate \( \int x^2 \, dx \), which results in \( \frac{x^3}{3} + C_2 \).

Solve for the Integration Constants

Combine the constants from both sides into a single constant. The equation becomes \( -e^{-y} = \frac{x^3}{3} + C \), where \( C = C_2 - C_1 \).

Solve for y

Solve for \( y \) in terms of \( x \) and \( C \). We have \( -e^{-y} = \frac{x^3}{3} + C \). Multiply through by \(-1\) to get \( e^{-y} = -\frac{x^3}{3} - C \). Taking the natural logarithm gives \( -y = \ln(-\frac{x^3}{3} - C) \). Then, multiply both sides by \(-1\) to isolate \( y \): \( y = -\ln(-\frac{x^3}{3} - C) \).

Express the General Solution

The general solution of the differential equation is \( y(x) = -\ln(-\frac{x^3}{3} - C) \), where \( C \) is the constant of integration.

Key concepts

Variable Separation

In differential equations, variable separation is a method used to simplify and solve equations, allowing the integration of both sides. It involves rearranging the differential equation to have all terms involving one variable on one side and all terms related to the other variable on the opposite side. This permits independent integration.

In the original exercise, we start with the equation: \( y' = e^y x^2 \), rewritten as \( \frac{dy}{dx} = e^y x^2 \). To separate variables, we would isolate terms involving \( y \) on one side and terms involving \( x \) on the other:- Move \( e^{-y} \) to the left: \( e^{-y} dy = x^2 dx \).- Ensure that each side is a function of only one variable before proceeding with integration.

Variable separation is quite powerful because once variables are separated, both sides of the equation can be integrated independently, paving the path for finding a solution to the differential equation.

Integration Constant

When solving differential equations by integration, you will encounter integration constants. These constants appear once we integrate a function and are crucial to computing the general solution.

During integration, any antiderivative of a function could be the solution, leading to a family of solutions. This uncertainty is represented by adding a constant, \( C \), to the result:

• The integration of \( \int e^{-y} \, dy \) results in \( -e^{-y} + C_1 \).
• The result for \( \int x^2 \, dx \) becomes \( \frac{x^3}{3} + C_2 \).

Often, both integration constants from each side can be combined into a single constant: \( C = C_2 - C_1 \).

This consolidated constant, \( C \), simplifies the solution and enables us to express the general solution of the differential equation. The constant of integration reflects the indefinite nature of integration and the multitude of solutions that can be obtained.

Logarithmic Integration

Logarithmic integration is necessary when dealing with integrals that have a form or transformation involving exponential functions. It comes into play particularly when solving differential equations involving exponentials, like in the given exercise.

After having separated the variables and integrated both sides, we needed to manipulate the expression involving \( e^{-y} \):- The integrated result, \( -e^{-y} = \frac{x^3}{3} + C \), is inverted.- By taking the natural logarithm, we find \( -y \): \( -y = \ln(-\frac{x^3}{3} - C) \).
Logarithmic functions provide a convenient way to express the solutions of equations where differentials or integrals involve exponential relationships.

This technique allows us to finally express \( y \) explicitly, offering us an identifiable solution form and helping simplify the presence of complex relationships.

General Solution

The general solution of a differential equation represents a family of solutions containing all possible specific solutions. This is where the variable and constant elements come together.

In the context of the original problem, we worked through separating variables, integrating, and addressing integration constants to arrive at the general solution:\[ y(x) = -\ln(-\frac{x^3}{3} - C) \] Here,

• \( y(x) \) denotes the function solution in terms of \( x \).
• \( C \) is the integration constant, embodying the potential different solutions.

The general solution is crucial because it provides the broadest scope of possible solutions for a differential equation and lets us apply boundary or initial conditions to find a particular solution tailored to specific circumstances.