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  4. The differential equation of all conics whose axes coincide with the axes of coordinates is of order

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Q. The differential equation of all conics whose axes coincide with the axes of coordinates is of order

Differential Equations

Solution:

$P(x, y)=a x^2+b y^2+2 h x y+2 g x+2 f y+c=0 $
$ a x+h y+g=0$
$ \frac{d P}{d x}=0 \Rightarrow 2 a x+2 h y+2 g=0 $
$ b y+h x+f=0 $
$ \frac{d P}{d y}=0 \Rightarrow 2 b y+2 h x+2 f=0 $
$ \Rightarrow h=g=f=0 $
conic : $ a x^2+b y^2+c=0$
$ \Rightarrow x^2+\left(\frac{b}{a}\right) y^2+\frac{c}{a}=0 $ order $=2$