Applications of Derivatives
Implicit Differentiation — Legendre-type ODE
nta_pyq_2024_apr
Grade 12
Question:
If $\log_e y=3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\dfrac{1}{2}$ is equal to:
$3e^{\pi/6}$
$9e^{\pi/2}$
$3e^{\pi/2}$
$9e^{\pi/6}$
Step-by-Step Solution
Key Concept: Differentiate $\ln y=3\sin^{-1}x$: $y'/y=3/\sqrt{1-x^2}$. Differentiate again to find $(1-x^2)y''-xy'=9y$.
$(1-x^2)y''-xy'=9y=9e^{\pi/2}$.
Correct Answer: 2