Class CBSE Class 12 Mathematics Differential Equations Q #811
KNOWLEDGE BASED
APPLY
1 Marks 2023 MCQ SINGLE
The general solution of the differential equation $x dy-(1+x^{2})dx=dx$ is:
(A) $y=2x+\frac{x^{3}}{3}+C$
(B) $y=2\log x+\frac{x^{3}}{3}+C$
(C) $y=\frac{x^{2}}{2}+C$
(D) $y=2\log x+\frac{x^{2}}{2}+C$
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Correct Answer: D
Explanation
$y=2\log x+\frac{x^{2}}{2}+C$

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Step-by-Step Solution

The given differential equation is: \(x \frac{dy}{dx} - (1+x^2) = 1\)

Rearrange the equation to isolate \(\frac{dy}{dx}\):

\(x \frac{dy}{dx} = 1 + 1 + x^2\)

\(x \frac{dy}{dx} = 2 + x^2\)

Separate the variables:

\(dy = \frac{2 + x^2}{x} dx\)

\(dy = (\frac{2}{x} + x) dx\)

Integrate both sides:

\(\int dy = \int (\frac{2}{x} + x) dx\)

\(y = 2 \int \frac{1}{x} dx + \int x dx\)

Perform the integration:

\(y = 2 \log|x| + \frac{x^2}{2} + C\)

Correct Answer: y=2\~log\~x+\frac{x^{2}}{2}+C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply the knowledge of differential equations and integration techniques to solve the given problem.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to solve the differential equation, including separation of variables and integration.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of standard methods for solving differential equations, a topic covered in the textbook.

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