Question

The integrating factor of the differential equation $2dy = (y + \cos x) dx$ is

• A. $e^{-2x}$
• B. $e^{-x/2}$
• C. $e^{2x}$
• D. $e^{x/2}$
• E. $-e^{x/2}$

Hint:

Always ensure the coefficient of $\frac{dy}{dx}$ is exactly 1 before identifying $P(x)$. Forgetting to divide by a constant coefficient like the '2' here is a very common mistake.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To find the integrating factor, we must first write the differential equation in the standard linear form: \(\frac{dy}{dx} + P(x)y = Q(x)\).
Step 2: Key Formula or Approach:
The integrating factor (I.F.) is given by \(e^{\int P(x) dx}\).
Step 3: Detailed Explanation:
Given equation:
\[ 2dy = (y + \cos x)dx \] Divide both sides by \(2dx\):
\[ \frac{dy}{dx} = \frac{y + \cos x}{2} \] \[ \frac{dy}{dx} = \frac{1}{2}y + \frac{1}{2}\cos x \] Rearrange to standard form \(\frac{dy}{dx} + P(x)y = Q(x)\):
\[ \frac{dy}{dx} - \frac{1}{2}y = \frac{1}{2}\cos x \] Identify \(P(x)\):
\[ P(x) = -\frac{1}{2} \] Calculate the integrating factor:
\[ \text{I.F.} = e^{\int P(x) dx} = e^{\int -\frac{1}{2} dx} \] \[ \text{I.F.} = e^{-\frac{1}{2}x} = e^{-x/2} \] Step 4: Final Answer:
The integrating factor is \(e^{-x / 2}\).