Problem 10 - Thomas Calculus

Question

Solve the differential equations. \(x \frac{d y}{d x}=\frac{\cos x}{x}-2 y, \quad x>0\)

Step-by-Step Solution

Answer

The solution is \( y = \frac{\sin x}{x^2} + \frac{C}{x^2} \).

Step 1: Rearrange the Equation

We start with the given equation: \( x \frac{d y}{d x} = \frac{\cos x}{x} - 2y \). This equation is in the form of a linear first-order differential equation. We want to rewrite it in standard linear form \( \frac{d y}{d x} + P(x) y = Q(x) \). Divide through by \(x\) to get: \( \frac{d y}{d x} + \frac{2}{x} y = \frac{\cos x}{x^2} \).

Step 2: Identify the Integrating Factor

For a linear differential equation \( \frac{d y}{d x} + P(x)y = Q(x) \), the integrating factor is given by \( \mu(x) = e^{\int P(x) \, dx} \). Here, \( P(x) = \frac{2}{x} \). Integrate \( P(x) \) to find: \( \int \frac{2}{x} \, dx = 2 \ln |x| = \ln x^2 \). Thus, the integrating factor is \( \mu(x) = e^{\ln x^2} = x^2 \).

Step 3: Multiply Through by the Integrating Factor

Multiply each term of the differential equation by the integrating factor \(x^2\): \( x^2 \frac{d y}{d x} + 2x y = \cos x \). This simplifies to \( \frac{d}{dx} (x^2 y) = \cos x \).

Step 4: Integrate Both Sides

Now integrate both sides of the equation: \( \int \frac{d}{dx} (x^2 y) \, dx = \int \cos x \, dx \). The left side integrates to \( x^2 y \). The integral of \( \cos x \) is \( \sin x \), so we have \( x^2 y = \sin x + C \), where \( C \) is the constant of integration.

Step 5: Solve for y

Finally, solve for \( y \) by dividing both sides by \( x^2 \): \( y = \frac{\sin x}{x^2} + \frac{C}{x^2} \). This is the general solution to the differential equation for \( x > 0 \).

Key Concepts

Linear First-Order Differential Equations

Linear first-order differential equations are a specific type of differential equations that involve functions and their first derivatives. They are called 'linear' because the function and its derivative appear to the first power.
These equations take the standard form:

• \( \frac{d y}{d x} + P(x) y = Q(x) \)

In this form, \( y \) is the unknown function of \( x \), \( \frac{d y}{d x} \) is the first derivative of \( y \) concerning \( x \), \( P(x) \) and \( Q(x) \) are known functions of \( x \).
Our task is to solve for \( y \). To achieve this, we first need to rearrange the given differential equation into this standard form by dividing or multiplying the equation as needed.

Integrating Factor

Once the equation is in standard linear form, an integrating factor is used. This is a clever mathematical tool that simplifies the process of solving the equation.
The integrating factor, denoted as \( \mu(x) \), is calculated using:

• \( \mu(x) = e^{\int P(x) \, dx} \)

This factor helps turn the left-hand side of the differential equation into a perfect derivative.
Once \( \mu(x) \) is found, we multiply every term in the equation by it. This transforms the equation into a form that can be integrated straightforwardly.
For example, in our exercise, the integrating factor was calculated as \( x^2 \). Multiplying by this factor simplified the solution process significantly.

General Solution

The general solution of a differential equation represents a family of solutions that includes every possible solution of the given differential equation. After simplifying the equation with the integrating factor, we integrate both sides to find the general solution.
For a first-order linear differential equation, the solution will usually include one arbitrary constant, denoted as \( C \).
Solving for \( y \), we have:

• \( y = \text{Particular Solution} + \frac{C}{\text{Transformed Factor}} \)

In our example, once we integrated, the solution was:

• \( y = \frac{\sin x}{x^2} + \frac{C}{x^2} \)

This equation implies a general family of solutions, illustrating how \( y \) behaves relative to \( x \), with arbitrarily chosen constants providing flexibility to fit initial conditions.