Problem 5 - Calculus Early Transcendentals

Question

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(2+2 y^{2}\right) y^{\prime}=e^{x} y $$

Step-by-Step Solution

Answer

The family of solutions is expressed implicitly: \( \frac{|y|}{\sqrt{1+y^2}} = Ce^{\frac{e^x}{4}} \).

Step 1: Identify Separation of Variables

The given differential equation is \( (2 + 2y^2) \frac{dy}{dx} = e^x y \). We will use separation of variables to solve it. Our goal is to separate the \( y \) terms from the \( x \) terms.

Step 2: Rearrange the Equation

To separate variables, divide both sides of the equation by \( y(2 + 2y^2) \) to isolate \( \frac{dy}{dx} \). This gives:\[ \frac{dy}{y(1 + y^2)} = \frac{e^x}{4}dx \].

Step 3: Integrate Both Sides

Integrate the left-hand side with respect to \( y \) and the right-hand side with respect to \( x \):\[ \int \frac{1}{y(1 + y^2)} \, dy = \int \frac{e^x}{4} \, dx \].

Step 4: Solve Each Integral

The left-hand side integral can be solved with partial fraction decomposition or recognizing it as:\[ \int \frac{1}{y} dy - \int \frac{y}{1+y^2} dy \].The first integral yields \( \ln |y| \), and the second, \( \frac{1}{2} \ln |1+y^2| \), using substitution. The right-hand side is straightforward:\[ \int \frac{e^x}{4} \, dx = \frac{e^x}{4} + C \].Thus we have:\[ \ln |y| - \frac{1}{2}\ln |1+y^2| = \frac{e^x}{4} + C \].

Step 5: Solve for y

Combine the logarithms using properties of logs:\[ \ln \left( \frac{|y|}{\sqrt{|1+y^2|}} \right) = \frac{e^x}{4} + C \].Exponentiate both sides:\[ \frac{|y|}{\sqrt{1+y^2}} = Ce^{\frac{e^x}{4}} \].Solve for \( y \) to get the explicit equation if possible.

Step 6: Express Family of Solutions Explicitly

From the equation \( \frac{|y|}{\sqrt{1+y^2}} = Ce^{\frac{e^x}{4}} \), solve for \( |y| \):\[ |y| = Ce^{\frac{e^x}{4}} \sqrt{1+y^2} \].Given the complexity, express the implicit form unless further simplification indicates a clear explicit expression is suitable.

Key Concepts

Separation of Variables

Differential equations often involve finding a function that satisfies a given equation involving derivatives. Separation of variables is a common method for solving first-order differential equations. This technique works by rearranging the equation such that all terms involving one variable, typically the dependent variable like \( y \), are on one side of the equation, and all terms involving the other variable, often the independent variable like \( x \), are on the other side.
This process breaks the problem into simpler, separate integrals. Once separated, you integrate each side with respect to its variable.

• Identify terms: Start by identifying which terms in the equation involve derivatives or functions of either \( y \) or \( x \).
• Rearrange: Make sure to rearrange the equation so that each side contains only one variable.
• Integrate: Integration can then proceed, often made simpler by having isolated each variable.

This approach simplifies the original problem into manageable parts, allowing you to solve each half of the equation separately. It's essential for students to practice identifying separable equations to streamline solving such differential equations.

Integration Techniques

Integration is the process of finding a function given its derivative. When working with differential equations, integrating is often a necessary step after separating variables. Various integration techniques can be applied depending on the complexity and form of the expression.
Some common methods include:

• Direct Integration: If the integral is straightforward, such as \( \int e^x \, dx = e^x + C \), integration is direct.
• Substitution: Useful when dealing with composite functions. For example, using \( u \)-substitution for \( \int y / (1 + y^2) \, dy \), letting \( u = 1 + y^2 \).

When solving our original equation, integration was vital in both the variable \( y \) and \( x \) sides. Direct integration applied to \( e^x \) while substitution helped solve the \( 1/(y(1 + y^2)) \).
Practicing various techniques helps in quickly identifying the easiest method to tackle an integral within differential equations.

Partial Fraction Decomposition

Partial fraction decomposition is particularly useful for integrating rational expressions. This technique involves expressing a complicated fraction as a sum of simpler fractions, making integration manageable. It's usually employed when you have a polynomial divided by another polynomial.
In our task, separating \( \frac{1}{y(1 + y^2)} \) into partial fractions or recognizing it as a convertible expression assists in integration. This involves:

• Breaking down the fraction: Decompose the original fraction into terms that are easier to integrate.
• Solving simpler integrals: After decomposition, integrate each term separately.

For example, \( \frac{1}{y(1 + y^2)} \) can be viewed as \( \frac{1}{y} - \frac{y}{1+y^2} \). Each term is then separately integrated. Being adept at partial fraction decomposition empowers you to transform complex fractions into formats easily integrated, which is essential when handling more complex differential equations.