Differential Equations
Differential Equation
nta_abhyas_2025
Grade 12
Question:
If the solution of the differential equation $y^2e^{x^2}dx + \sin(x^2)y'dx = \frac{5}{6}dx\,2\sin(x^2)y'e = x^2 + C$ (where $C$ is an arbitrary constant), then the value of $k$ is equal to
Step-by-Step Solution
Key Concept: Recognize exact differential equations and integrate appropriately
Rewrite the given equation as $d\left(\sin(x^3)\right) y^3 + \sin(x^3) dx = idx$. This can be written as $d(\sin(x^3)) = idx$. On integrating, we get $\sin(x^3) y^3 = \frac{x^2}{2} + C$, giving $k = 3$.
Correct Answer: 3