Question

The differential equation of all conics with centre at origin is of order

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

The correct option is B $3$

Equation of conic with centre $(0,0)$ is

$a{x}^2+2hxy+b{y}^2=1$ ...(i)

$2ax+2hy+2hx{y}^1+2by{y}^1=0$

$\Rightarrow ax+h(y+x{y}^1)+by{y}^1=0$ ...(ii)

multiplying $x^1$

$a{x}^2+h(xy+x^2{y}^1)+bxy{y}^1=0$ ...(iii)

(iii)-(ii)

$h(x^2{y}^1-xy)+b(xy{y}^1-y^2)=-1$

$\Rightarrow y(hx+by)-xy(hx+by)=1$

$\Rightarrow (y-x{y}^1)(hx+by)=1$

$\therefore hx+by=\frac{1}{y-x{y}^{1}b}$

Diff wrt ${}^{\prime }{x}^{\prime }$

$h+b{y}^1=\frac{x{y}_2}{(y-x{y}_1{)}^2}$

$b(y-x{y}_1)=\frac{y-x{y}_1-x^2{y}_1}{(y-x{y}_2{)}^2}$

$b=\frac{y-x{y}_1-x^2{x}_1}{(y-x{y}_1{)}^2}$

Differential equation

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

$\therefore$ order is $3$