Question

Form a differential equation for the family of curves represented by $a{x}^2+b{y}^2=1$ , where a & b are arbitary constants.

• A. $xy\frac{d^{2}y}{d{x}^2}+y{\left(\frac{dy}{dx}\right)}^2-x\frac{dy}{dx}=0$
• B. $xy\frac{d^{2}y}{d{x}^2}-x{\left(\frac{dy}{dx}\right)}^2+y\frac{dy}{dx}=0$
• C. $xy\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-y\frac{dy}{dx}=0$
• D. $xy\frac{d^{2}y}{d{x}^2}-y{\left(\frac{dy}{dx}\right)}^2+x\frac{dy}{dx}=0$

Answer 1

The correct option is C $xy\frac{d^{2}y}{d{x}^2}+x{\left(\frac{dy}{dx}\right)}^2-y\frac{dy}{dx}=0$

Given $a{x}^2+b{y}^2=1$

Differntiate wrt x, we get

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

$\Rightarrow ax=-by\frac{dy}{dx}$------(1)

differentiate wrt x, we get

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

$Divide\left(1\right)by\left(2\right)$

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

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