Problem 1 - Technical Mathematics with Calculus

Question

Find the general solution to each differential equation. $$(x-y) d x-2 x d y=0$$

Step-by-Step Solution

Answer

The general solution to the differential equation is \(y = 2x + \frac{C}{x}\), where \(C\) is an arbitrary constant.

Step 1: Identify the Type of Differential Equation

The given differential equation is a first-order linear differential equation of the form \(y' + P(x)y = Q(x)\), where \(P(x) = \frac{1}{x}\) and \(Q(x) = 4\). To solve it, we will use an integrating factor.

Step 2: Find the Integrating Factor

The integrating factor, \(\mu(x)\), is given by \(\mu(x) = e^{\int P(x) dx}\). We calculate the integral of \(P(x)\) to find \(\mu(x)\). \[\mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = |x|\]. Since \(x > 0\) for the logarithm to be defined, we can drop the absolute value and use \(\mu(x) = x\).

Step 3: Multiply Through by the Integrating Factor

Now multiply every term of the differential equation by the integrating factor \(\mu(x) = x\) to get \[x y' + y = 4x\].

Step 4: Write the Left-Hand Side as a Derivative

Recognize that the left-hand side is the derivative of the product of the integrating factor and the function \(y\): \[(xy)' = x y' + y\].

Step 5: Integrate Both Sides with Respect to \(x\)

Integrate both sides with respect to \(x\) to find \[\int (xy)' dx = \int 4x dx\], which simplifies to \[xy = 2x^2 + C\], where \(C\) is the constant of integration.

Step 6: Solve for \(y\)

Now, divide both sides of \(xy = 2x^2 + C\) by \(x\) (assuming \(x eq 0\)) to solve for \(y\): \[y = 2x + \frac{C}{x}\].

Key Concepts

Integrating Factor Method

The integrating factor method is a powerful tool for solving first-order linear differential equations, which commonly appear in various fields of science and engineering. This method transforms a non-exact differential equation into an exact one by multiplying each term by a specially chosen function, called the integrating factor.

The key to the integrating factor method lies in the formula \( \mu(x) = e^{\int P(x) dx} \), which involves an exponential function raised to the integral of the function P(x) from the standard linear form of the equation \( y' + P(x)y = Q(x) \).

Once the integrating factor \( \mu(x) \) is determined, we multiply the entire differential equation by this factor, which enables us to write the left-hand side of the equation as a derivative of the product \( \mu(x)y \). This simplifies integration because we can apply the reverse of the product rule for differentiation. After integrating both sides with respect to \( x \), the solution emerges with the function \( y \) expressed in terms of \( x \) and a constant of integration.

This method requires a clear understanding of derivatives and integrals, and it's essential to practice these calculations to gain proficiency.

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They play a crucial role in modeling situations where change is constant; for example, they can describe how populations grow, how heat moves, and how springs oscillate. These equations arise whenever a relationship involves rates of change or quantities varying over time.

First-order linear differential equations are a subclass that have the general form \( y' + P(x)y = Q(x) \), where \( y' \) represents the derivative of \( y \) with respect to \( x \), and \( P(x) \) and \( Q(x) \) are functions of \( x \) only.

Understanding the behavior and solutions of such equations aids in predicting and controlling systems in the real world. Solutions can be general, involving arbitrary constants, or particular, where specific conditions are applied. In the exercise provided, we focused on finding the general solution, which gives us the family of all possible solutions to the given differential equation.

Constant of Integration

The constant of integration, symbolized by \( C \), is a fundamental aspect when integrating functions in calculus. It represents an arbitrary constant that is added to the function as a result of the indefinite integration process.

An indefinite integral, unlike a definite integral, does not have specified limits of integration. Therefore, it incorporates an infinite number of potential constants. Since differentiation erases any added constants (as their derivative is zero), when we find the antiderivative of a function, we must include \( C \) to account for all those possibilities.

In the context of solving differential equations, the constant of integration is key to forming the general solution, allowing the solution to fit a wide range of initial conditions or boundary values. In the provided exercise, after integrating both sides, we include \( C \) in our solution to represent the infinite number of solutions that satisfy our differential equation, covering all possible scenarios that align with the mathematical model.