Question

if $y^2=ax+bx+c$, then $y^3\frac{d^{2}y}{d{x}^2}$ is

• A. a constant
• B. a function of $x$ only
• C. a function of $y$ only
• D. a function of $x$ and $y$

Answer 1

Answer: (A)

a constant

Given $y^2=ax+bx=x$

differentiate w.r.t. to x

$2y\cdot \frac{dy}{dx}=a+b$ …….$\left(1\right)$

Again differentiate wrt to x

$2{\left(\frac{dy}{dx}\right)}^2+2y\cdot \frac{d^{2}y}{d{x}^2}=0$

${\left(\frac{dy}{dx}\right)}^2+y\frac{d^{2}y}{d{x}^2}=0$

$y\frac{d^{2}y}{d{x}^2}=-{\left(\frac{dy}{dx}\right)}^2$

Multiplying both side by $y^2$

$y^3\frac{d^{2}y}{d{x}^2}=-y^2{\left(\frac{dy}{dx}\right)}^2$

From equation $\left(1\right)$

$y^3\frac{d^{2}y}{d{x}^2}=-y^2{\left(\frac{a+b}{2y}\right)}^2$

$y^3\frac{d^{2}y}{d{x}^2}=-\frac{(a+b{)}^2}{4}$

So, $y^3\frac{d^{2}y}{d{x}^2}$ is a constant.

$\therefore$ Option A is correct.