Differential Equations
Variable Separable — Nested Radical Substitution
nta_pyq_2026_jan
Grade 12
Question:
If $y=y(x)$ satisfies the differential equation $16\!\left(\sqrt{x+9\sqrt{x}}\right)\!\left(4+\sqrt{9+\sqrt{x}}\right)\cos y\,dy=\left(1+2\sin y\right)dx$, $x>0$ and $y(256)=\dfrac{\pi}{2}$, $y(49)=\alpha$, then $2\sin\alpha$ is equal to:
2(\sqrt{2}-1)
3(\sqrt{2}-1)
2\sqrt{2}-1
\sqrt{2}-1
Step-by-Step Solution
Key Concept: Separate variables: $\dfrac{\cos y}{1+2\sin y}dy=\dfrac{dx}{16\sqrt{x+9\sqrt{x}}(4+\sqrt{9+\sqrt{x}})}$. Let $t=4+\sqrt{9+\sqrt{x}}$ to simplify RHS to $\tfrac{dt}{4t}$.
$2\sin\alpha=2\sqrt{2}-1$.
Correct Answer: 3