The curve $C$ has equation $y^3 + 2x^2 y = 1$ and it cuts the $y$-axis at point $N$ . Show t

Question

The curve $C$ has equation $y^3 + 2x^2 y = 1$ and it cuts the $y$-axis at point $N$ . Show that $\left(\alpha y^2 + \beta x^2\right)\frac{d^2 y}{dx^2} + \mu y\left(\frac{dy}{dx}\right)^2 + \lambda x\frac{dy}{dx} + \gamma y = 0$ where $\alpha, \beta, \mu, \lambda$ and $\gamma$ are constants to be determined. [3] Find the values of $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$ at $N$ and comment on the nature of $N$ .[3]

Accepted Answer

(a) $\alpha=3, \beta=2, \mu=6, \lambda=8, \gamma=4$ (b) $N=(0,1)$, $\dfrac{dy}{dx}=0$, $\dfrac{d^2y}{dx^2}=-\dfrac43$; maximum

2025 HCI Promo Q2

2025 HCI Promo Q2

Solution:

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