Class CBSE Class 12 Mathematics Differential Equations Q #1345
KNOWLEDGE BASED
REMEMBER
3 Marks 2024 AISSCE(Board Exam) SA
Find the particular solution of the differential equation $\frac{dy}{dx}-2xy=3x^{2}e^{x^{2}};y(0)=5.$
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Detailed Solution

Step 1: Identify the type of differential equation

The given differential equation is of the form $\frac{dy}{dx} + P(x)y = Q(x)$, which is a first-order linear differential equation.

Step 2: Find the integrating factor (IF)

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

Step 3: Multiply the differential equation by the integrating factor

Multiplying the given differential equation by the integrating factor $e^{-x^2}$, we get: $e^{-x^2}\frac{dy}{dx} - 2xe^{-x^2}y = 3x^2e^{x^2}e^{-x^2}$ $e^{-x^2}\frac{dy}{dx} - 2xe^{-x^2}y = 3x^2$

Step 4: Integrate both sides with respect to x

The left-hand side is the derivative of $y \cdot IF$ with respect to $x$. So, we have: $\int \frac{d}{dx}(ye^{-x^2}) dx = \int 3x^2 dx$ $ye^{-x^2} = \int 3x^2 dx$ $ye^{-x^2} = x^3 + C$, where $C$ is the constant of integration.

Step 5: Find the general solution

Multiply both sides by $e^{x^2}$ to get the general solution: $y = e^{x^2}(x^3 + C)$

Step 6: Apply the initial condition y(0) = 5

Given $y(0) = 5$, substitute $x = 0$ and $y = 5$ into the general solution: $5 = e^{0^2}(0^3 + C)$ $5 = 1(0 + C)$ $C = 5$

Step 7: Find the particular solution

Substitute $C = 5$ into the general solution: $y = e^{x^2}(x^3 + 5)$

Final Answer: $y = e^{x^2}(x^3 + 5)$

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the method to solve a first-order linear differential equation and applying the formula for the integrating factor.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (solving a first-order linear differential equation) to arrive at the solution. This involves knowing the steps and applying them in the correct order.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's ability to solve differential equations, a standard topic in the syllabus. The question is a straightforward application of a known method.

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