Question

Find the differential equation associated with the primitive $x^2+y^2+2ax+2by+c=0,$ where a,b,c are arbitrary constants.

• A. $\frac{d^{3}y}{d{x}^3}[1+{\left(\frac{dy}{dx}\right)}^2]=3\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
• B. $\frac{d^{3}y}{d{x}^3}[1-{\left(\frac{dy}{dx}\right)}^2]=3\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
• C. $\frac{d^{3}y}{d{x}^3}[1+{\left(\frac{dy}{dx}\right)}^2]=4\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
• D. $\frac{d^{3}y}{d{x}^3}[1+{\left(\frac{dy}{dx}\right)}^2]=2\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

Answer 1

The correct option is A $\frac{d^{3}y}{d{x}^3}[1+{\left(\frac{dy}{dx}\right)}^2]=3\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

$x^2+y^2+2ax+2by+c=0\cdots \left(1\right)$
Differentiate w.r.t $x$
$2(x+y{y}_1)+2(a+b{y}_1)=0$
or $x+y{y}_1+(a+b{y}_1)=0\cdots \left(2\right)$
Again
differentiate w.r.t. $x$
$1+y{y}_2+y_1^2+b{y}_2=0\cdots \left(3\right)$
Differentiate (3) w.r.t. $x$
$y{y}_3+y_1{y}_2+2y_1{y}_1+b{y}_3=0\cdots \left(4\right)$
Eliminating b between (3) and (4), we get $\frac{1+y{y}_2+y_1^2}{y{y}_3+3y_1{y}_2}=\frac{-b{y}_2}{-b{y}_3}=\frac{y_2}{y_3}$

or $y_3(1+y{y}_2+y_1^2)=y_2(y{y}_3+3y_1{y}_2)$
Cancel the terms of $y{y}_2{y}_3$
$\therefore y_3(1+y_1^2)=3y_1{y}_2^2$
$\Rightarrow \frac{d^{3}y}{d{x}^3}[1+{\left(\frac{dy}{dx}\right)}^2]=3\frac{dy}{dx}{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$