Question

Use the method of separation of variables to find a general solution to the differential equation \(y^{\prime}=2 x y+3 y-4 x-6\)

Short answer

Separate the equation as \( \frac{dy}{dx} = y (2x + 3) - (4x + 6) \), integrate both sides, and solve for \( y \).

Step-by-step solution

Rewrite the Equation in Separable Form

To use the method of separation of variables, first rearrange the given differential equation \( y' = 2xy + 3y - 4x - 6 \). Begin by factoring \( y \) from the terms that contain it: \( y' = y(2x + 3) - (4x + 6) \). This form will allow us to separate variables.

Separate Variables

Separate the variables by moving all terms involving \(y\) to one side and all terms involving \(x\) to the other side. Let \( z = y' \) and express the equation as: \( \frac{dy}{dx} = y(2x + 3) - (4x + 6) \). Separate by rearranging: \( \frac{dy}{y(2x + 3) - (4x + 6)} = dx \).

Integrate Both Sides

Integrate both sides of the separated equation. On the left side, we have an integral with respect to \( y \) and, on the right side, an integral with respect to \( x \). The integration will involve manipulating the expression to simplify it before integrating. This can involve partial fraction decomposition if necessary to simplify the integration process.

Evaluate the Integrals

After setting up the integrals, solve them to find \( y \) in terms of \( x \). Perform necessary algebraic manipulations and partial fractions if needed: \( \int \frac{1}{y(2x + 3) - (4x + 6)} \, dy = \int \, dx \). Simplify both integrals to solve for \( y \).

Solve for the General Solution

After integration, you will have two expressions, one involving \( y \) and another involving \( x \). Combine these to write the general solution for the differential equation. This will typically include a constant of integration \( C \), indicating the general solution form.

Key concepts

Differential Equations

Differential equations involve relationships between functions and their derivatives. These equations describe how a function changes along with its independent variable. In our original exercise, we have the differential equation: \[y^{\prime} = 2xy + 3y - 4x - 6\] The primary task here is to solve for \(y\), the dependent variable, in terms of \(x\), the independent variable. Solving differential equations like this typically gives us insights into the dynamics or rates of change in various systems, such as physics, biology, or economics. The process often begins with rearranging and setting up the equation into a form that can be more easily tackled by methods like separation of variables or integration techniques.

Integral Calculus

Integral calculus offers a powerful set of tools for solving differential equations by integrating functions and finding anti-derivatives. In the context of our exercise, once we've rearranged the differential equation into a separable form, the next step is to integrate. Here's where integral calculus shines:

• It allows us to integrate both sides of the equation, separating the variable on each side.
• The left side involves integrating with respect to \(y\), while the right side does so with respect to \(x\).

By performing integrations, we can find a function or expression that satisfies the original differential equation, essentially "solving" the equation. This solution can sometimes involve additional techniques like partial fraction decomposition to make the integrals manageable.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler parts. This method is particularly helpful when solving integrals that arise in the process of separating variables in differential equations. When we separated variables in our exercise, we ended up with an expression that could potentially be simplified using partial fractions: \[\int \frac{1}{y(2x + 3) - (4x + 6)} \, dy = \int \, dx\] The goal here is to express the more complicated fraction as a sum of simpler fractions that are easier to integrate. This step often involves recognizing patterns or using algebraic techniques to manipulate the expression. Once decomposed, each simple fraction can be integrated individually.The result from partial fraction decomposition is key to solving the integral part of the differential equation, ultimately leading us to the general solution.