Question

Integrating factor of the differential equation $(1 - x^2)\frac{dy}{dx} - xy = 1$ is

• A. $1 - x^2$
• B. $\frac{1}{2} \log (1 - x^2)$
• C. $\frac{x}{1 - x^2}$
• D. $\sqrt{1 - x^2}$

Hint:

Always remember to bring the differential equation to the standard form $\frac{dy}{dx} + P(x)y = Q(x)$ before identifying $P(x)$. Forgetting to divide by the coefficient of $\frac{dy}{dx}$ is a very common mistake.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To find the integrating factor of a linear differential equation, it must first be written in the standard form: $\frac{dy}{dx} + P(x)y = Q(x)$. The integrating factor (I.F.) is then given by the formula $\text{I.F.} = e^{\int P(x) \, dx}$.

Step 2: Key Formula or Approach:
1. Rearrange the given equation into standard linear form by dividing the entire equation by the coefficient of $\frac{dy}{dx}$.
2. Identify the function $P(x)$.
3. Compute the integral $\int P(x) \, dx$.
4. Calculate the integrating factor as $e^{\int P(x) \, dx}$.

Step 3: Detailed Explanation:
The given differential equation is: \[ (1 - x^2)\frac{dy}{dx} - xy = 1 \] Divide the entire equation by $(1 - x^2)$ to bring it to standard form: \[ \frac{dy}{dx} - \frac{x}{1 - x^2}y = \frac{1}{1 - x^2} \] Comparing this with the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$, we identify: \[ P(x) = -\frac{x}{1 - x^2} \] Now, we calculate the integral of $P(x)$: \[ \int P(x) \, dx = \int -\frac{x}{1 - x^2} \, dx \] Let $1 - x^2 = t$. Then, differentiating both sides with respect to $x$ gives $-2x \, dx = dt$, which implies $-x \, dx = \frac{dt}{2}$. Substituting these into the integral: \[ \int -\frac{x}{1 - x^2} \, dx = \int \frac{1}{t} \cdot \frac{dt}{2} = \frac{1}{2} \int \frac{1}{t} \, dt \] \[ = \frac{1}{2} \log |t| = \frac{1}{2} \log(1 - x^2) = \log((1 - x^2)^{1/2}) = \log(\sqrt{1 - x^2}) \] Finally, the integrating factor is: \[ \text{I.F.} = e^{\int P(x) \, dx} = e^{\log(\sqrt{1 - x^2})} = \sqrt{1 - x^2} \]
Step 4: Final Answer:
The integrating factor is $\sqrt{1 - x^2}$.