Finding the Integrating Factor of a Differential Equation

We are asked to find the integrating factor for the given first-order linear differential equation.

The given differential equation is:

\( x \frac{dy}{dx} - 2y = x^3 \)

To find the integrating factor, we first need to write the differential equation in the standard form of a first-order linear differential equation, which is:

\( \frac{dy}{dx} + P(x)y = Q(x) \)

We need to manipulate the given equation to match this standard form. We can do this by dividing the entire equation by \(x\) (assuming \(x \neq 0\)):

\( \frac{x \frac{dy}{dx}}{x} - \frac{2y}{x} = \frac{x^3}{x} \)

This simplifies to:

\( \frac{dy}{dx} - \frac{2}{x}y = x^2 \)

Now, comparing this equation with the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \), we can identify \(P(x)\) and \(Q(x)\).

• \( P(x) = -\frac{2}{x} \)
• \( Q(x) = x^2 \)

The integrating factor (IF) for a first-order linear differential equation in standard form is given by the formula:

\( IF = e^{\int P(x) dx} \)

Now we need to calculate the integral of \(P(x)\):

\( \int P(x) dx = \int \left(-\frac{2}{x}\right) dx \)

\( \int P(x) dx = -2 \int \frac{1}{x} dx \)

The integral of \( \frac{1}{x} \) with respect to \(x\) is \( \ln|x| \). So,

\( \int P(x) dx = -2 \ln|x| \)

Now, substitute this result back into the formula for the integrating factor:

\( IF = e^{\int P(x) dx} = e^{-2 \ln|x|} \)

We can use properties of logarithms and exponentials to simplify this expression. Recall that \( a \ln b = \ln b^a \) and \( e^{\ln c} = c \).

First, use the property \( a \ln b = \ln b^a \):

\( e^{-2 \ln|x|} = e^{\ln(|x|^{-2})} \)

Next, use the property \( e^{\ln c} = c \):

\( e^{\ln(|x|^{-2})} = |x|^{-2} \)

Since \( |x|^2 = x^2 \), we have \( |x|^{-2} = \frac{1}{|x|^2} = \frac{1}{x^2} \).

Therefore, the integrating factor is:

\( IF = \frac{1}{x^2} \)

Comparing this result with the given options, we find that it matches one of them.

Revision Table: Key Concepts for Integrating Factor

Additional Information on Linear Differential Equations and Integrating Factors

A linear first-order differential equation is one of the simplest types of differential equations that can be solved analytically. The integrating factor method is a standard technique used specifically for this type of equation.

Here's a brief overview of the steps to solve a linear first-order ODE using the integrating factor method:

1. Write in Standard Form: Ensure the equation is in the form \( \frac{dy}{dx} + P(x)y = Q(x) \). If not, divide by the coefficient of \( \frac{dy}{dx} \).
2. Find \( P(x) \): Identify the function \(P(x)\) which is the coefficient of \(y\).
3. Calculate the Integrating Factor: Compute \( IF = e^{\int P(x) dx} \). Remember to include the constant of integration for the integral, but it's usually omitted when calculating the IF itself as any constant multiple will still serve as an integrating factor.
4. Multiply by the IF: Multiply the standard form equation by the integrating factor. The left side should become the derivative of the product \( y \cdot IF \). That is, \( IF \left(\frac{dy}{dx} + P(x)y\right) = \frac{d}{dx}(y \cdot IF) \).
5. Integrate Both Sides: The equation becomes \( \frac{d}{dx}(y \cdot IF) = Q(x) \cdot IF \). Integrate both sides with respect to \(x\): \( \int \frac{d}{dx}(y \cdot IF) dx = \int Q(x) \cdot IF dx \). This gives \( y \cdot IF = \int Q(x) \cdot IF dx + C \), where \(C\) is the constant of integration for the differential equation itself.
6. Solve for \(y\): Finally, divide by the integrating factor to get the general solution for \(y(x)\): \( y = \frac{1}{IF} \left( \int Q(x) \cdot IF dx + C \right) \).

The integrating factor is crucial because it transforms the left side of the linear equation into a form that is easily integrated, simplifying the solution process significantly.