Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
The curve passing through the point $(1, 1)$ satisfies the differential equation $\frac{dy}{dx} + \frac{\sqrt{(x^2-1)(y^2-1)}}{xy} = 0$. If the curves passes through the point $\left(\sqrt{2}, k\right)$, then the value of $[k]$ is (where $[.]$ represents greatest integer function).
Step-by-Step Solution
Key Concept: Separate variables and use substitution $y^2 - 1 = t^2$ to convert the integrals into manageable forms.
Given $\frac{dy}{dx} = \frac{\sqrt{(x^2-1)(y^2-1)}}{xy}$, separate variables: $\int \frac{y}{\sqrt{y^2-1}} dy = -\int \frac{\sqrt{x^2-1}}{x} dx$. Let $y^2 - 1 = t^2$, so $2y\,dy = 2t\,dt$. Then $\int t\,dt = -\int \frac{\sqrt{x^2-1}}{x} dx$, which integrates to find the implicit solution relating $x$ and $y$.
Correct Answer: 1