Differential Equations
Bernoulli/Substitution DE
nta_pyq_2024_apr
Grade 12
Question:
Let $y=y(x)$ be the solution curve of the differential equation $\sec y\dfrac{dy}{dx}+2x\sin y=x^3\cos y$, $y(1)=0$. Then $y(\sqrt{3})$ is equal to:
$\dfrac{\pi}{3}$
$\dfrac{\pi}{6}$
$\dfrac{\pi}{12}$
$\dfrac{\pi}{4}$
Step-by-Step Solution
Key Concept: Multiply by $\sec y$: $\sec^2y\frac{dy}{dx}+2x\tan y=x^3$. Let $t=\tan y$: $\frac{dt}{dx}+2xt=x^3$. IF $=e^{x^2}$.
$\tan y=(x^2-1)/2$. $y(\sqrt{3})=\pi/4$.
Correct Answer: 4