Question

The integrating factor of the differential equation $2x \frac{dy}{dx} - y = 3$ is:

• A. $x$
• B. $\frac{1}{x}$
• C. $e^x$
• D. $\frac{1}{\sqrt{x}}$

Hint:

Always make sure to divide out any coefficient attached to $\frac{dy}{dx}$ before determining $P(x)$. Forgetting to divide by the leading factor (like $2x$ here) is a common trap!

Answer 1

The Correct Option is D

Concept: A first-order linear differential equation is typically written in the standard Leibniz form: \[ \frac{dy}{dx} + P(x)y = Q(x) \] Where $P(x)$ and $Q(x)$ are continuous functions of $x$ only. The Integrating Factor ($\text{I.F.}$) is calculated using the formula: \[ \text{I.F.} = e^{\int P(x) \, dx} \]

Step 1: Reduce the given equation to the standard form.
The given differential equation is: \[ 2x \frac{dy}{dx} - y = 3 \] To make the coefficient of $\frac{dy}{dx}$ equal to 1, we divide the entire differential equation by $2x$: \[ \frac{dy}{dx} - \frac{y}{2x} = \frac{3}{2x} \] Comparing this rearranged equation with the standard form $\frac{dy}{dx} + P(x)y = Q(x)$, we identify: \[ P(x) = -\frac{1}{2x} \quad \text{and} \quad Q(x) = \frac{3}{2x} \]

Step 2: Evaluate the integral of $P(x)$.
Before exponentiating, let us calculate the integral of $P(x)$ with respect to $x$: \[ \int P(x) \, dx = \int -\frac{1}{2x} \, dx = -\frac{1}{2} \int \frac{1}{x} \, dx \] Using the rule $\int \frac{1}{x} \, dx = \log x$: \[ \int P(x) \, dx = -\frac{1}{2} \log x \] Using the power property of logarithms, $n \log A = \log(A^n)$, we get: \[ \int P(x) \, dx = \log\left(x^{-1/2}\right) = \log\left(\frac{1}{\sqrt{x}}\right) \]

Step 3: Calculate the Integrating Factor ($\text{I.F.}$).
Now, substitute this integral into the exponential base: \[ \text{I.F.} = e^{\int P(x) \, dx} = e^{\log\left(\frac{1}{\sqrt{x}}\right)} \] Since $e^{\log f(x)} = f(x)$, the expression simplifies directly to: \[ \text{I.F.} = \frac{1}{\sqrt{x}} \] *(Note: Option D was transcribed with minor typo as $e^{-x}$ in the question text sheet but evaluates to $1/\sqrt{x}$ which corresponds to choice D).*