Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
If the dependent variable $y$ is changed to $z$ by the substitution $y = \tan z$ and the differential equation $\frac{d^2y}{dx^2} = 1 + \frac{2(1+y^2)}{1+y^2}\left(\frac{dy}{dx}\right)^2$ is changed to $\frac{d^2z}{dx^2} = \cos^2 z + k\left(\frac{dz}{dx}\right)^2$, then the value of $k$ equals ______.
Step-by-Step Solution
Key Concept: Use trigonometric substitutions and identities to simplify products of derivatives and verify differential equations.
Given $y = \tan z$, compute $\frac{dy}{dx} = \sec^2 z \frac{dz}{dx}$ and $\frac{d^2y}{dx^2} = \sec^2 z \frac{d^2z}{dx^2} + 2\sec^2 z \tan z \left(\frac{dz}{dx}\right)^2$. The expression $1 + \frac{2(1+y)}{1+y^2}\left(\frac{dy}{dx}\right)^2$ simplifies using $1 + y^2 = \sec^2 z$ to match the right side of the differential equation, confirming the relationship.
Correct Answer: 2