Question:
The expression satisfying the differential equation
\[
(x^2 - 1) \frac{dy}{dx} + 2xy = 1
\]
is:
The expression satisfying the differential equation
\[
(x^2 - 1) \frac{dy}{dx} + 2xy = 1
\]
is:
Show Hint
Put the DE in linear form before finding the IF.
Updated On: Mar 23, 2026
- \(x^2y-xy^2=c\)
- \((y^2-1)x=y+c\)
- \((x^2-1)y=x+c\)
- none of these
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Write as linear in \(y\):
\[ \frac{dy}{dx} + \frac{2x}{x^2 - 1} y = \frac{1}{x^2 - 1} \]
Step 2: Integrating factor:
\[ \text{IF} = e^{\int \frac{2x}{x^2 - 1} dx} = x^2 - 1 \]
Step 3: Hence:
\[ (x^2 - 1) y = \int 1 \, dx = x + c \]
\[ \frac{dy}{dx} + \frac{2x}{x^2 - 1} y = \frac{1}{x^2 - 1} \]
Step 2: Integrating factor:
\[ \text{IF} = e^{\int \frac{2x}{x^2 - 1} dx} = x^2 - 1 \]
Step 3: Hence:
\[ (x^2 - 1) y = \int 1 \, dx = x + c \]
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