Question

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

• A. $e^{2x}$
• B. $-\frac{1}{2}e^{-2x}$
• C. $\frac{1}{2}e^{-2x}$
• D. $\frac{1}{2}e^{-2x}$ (Duplicate)
• E. $e^{-2x}$

Hint:

I.F. $= e^{\int P dx}$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
The given differential equation is a first-order linear differential equation. An equation of this type has the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\). To solve such an equation, we first find an integrating factor (I.F.).
Step 2: Key Formula or Approach
The integrating factor for a linear differential equation in the form \(\frac{dy}{dx} + P(x)y = Q(x)\) is given by the formula:
\[ \text{I.F.} = e^{\int P(x) dx} \] Step 3: Detailed Explanation
1. Identify P(x) from the given equation.
The differential equation is \(\frac{dy}{dx} - 2y = 2x - 3\).
Comparing this with the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\), we can identify:
\[ P(x) = -2 \] \[ Q(x) = 2x - 3 \] 2. Calculate the integral of P(x).
\[ \int P(x) dx = \int -2 \, dx = -2x \] (We do not need to add the constant of integration when finding the integrating factor).
3. Calculate the integrating factor.
Using the formula I.F. = \(e^{\int P(x) dx}\):
\[ \text{I.F.} = e^{-2x} \] Step 4: Final Answer
The integrating factor of the differential equation is \(e^{-2x}\).