Question

If $y^2$ = $a{x}^2$ + $bx$ + $c$ , 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: (B)

a function of x only

$y^2=a{x}^2+bx+c$

$\Rightarrow 2y\frac{dy}{dx}=\frac{d}{dx}\left(a{x}^2\right)+\frac{d}{dx}\left(bx\right)+\frac{d}{dx}\left(c\right)$

$\Rightarrow 2y\frac{dy}{dx}=2ax+b+0$

$\Rightarrow 2\frac{d}{dx}\left(y\frac{dy}{dx}\right)=\frac{d}{dx}(2ax+b)$

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

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

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

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

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

$=(a{x}^2+bx+c)(a-4a^2{x}^2-b^2-4axb)$ is a function of $x$ only