Class CBSE Class 12 Mathematics Differential Equations Q #1385
KNOWLEDGE BASED
UNDERSTAND
3 Marks 2025 AISSCE(Board Exam) SA
Solve the following differential equation: $(1+x^{2})\frac{dy}{dx}+2xy=4x^{2}$.
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Detailed Solution

Step 1: Rewrite the differential equation in standard form

Divide the entire equation by $(1+x^2)$ to get the standard form of a first-order linear differential equation:

Step 2: Identify P(x) and Q(x)

Comparing the equation $\frac{dy}{dx} + \frac{2x}{1+x^2}y = \frac{4x^2}{1+x^2}$ with the standard form $\frac{dy}{dx} + P(x)y = Q(x)$, we have:

Step 3: Calculate the Integrating Factor (IF)

The integrating factor is given by $IF = e^{\int P(x) dx}$. In this case, $P(x) = \frac{2x}{1+x^2}$. So,

Step 4: Evaluate the integral in the exponent

To find $\int \frac{2x}{1+x^2} dx$, let $u = 1+x^2$, then $du = 2x dx$. Thus,

Step 5: Determine the Integrating Factor

Now we can find the integrating factor:

Step 6: Write the general solution

The general solution of the differential equation is given by:

Step 7: Substitute the values of IF and Q(x)

Substitute $IF = (1+x^2)$ and $Q(x) = \frac{4x^2}{1+x^2}$ into the general solution:

Step 8: Simplify and integrate

Simplify the integral:

Step 9: Evaluate the integral

The integral of $4x^2$ is $\frac{4}{3}x^3$. Therefore,

Step 10: Solve for y

Divide both sides by $(1+x^2)$ to solve for $y$:

Final Answer: $y = \frac{4x^3}{3(1+x^2)} + \frac{C}{1+x^2}$

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Pedagogical Audit
Bloom's Analysis: This is an UNDERSTAND question because the student needs to understand the concept of differential equations, integrating factors, and how to apply them 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 finding the integrating factor and applying the formula for the general solution.<\/span>
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's knowledge of solving linear differential equations, a standard topic in the syllabus.

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