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  4. The differential equation of all parabolas having their axis of symmetry coinciding with the axis of x has its order and degree respectively

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Q. The differential equation of all parabolas having their axis of symmetry coinciding with the axis of $x$ has its order and degree respectively

Differential Equations

Solution:

$\text { equation }( x - a )^2+ y ^2=( x - b )^2 \quad[ S =( a , 0) ; D : x = b ] $
$y ^2=\left( b ^2- a ^2\right)+2 x ( a - b )$
differentiate twice to get $y \frac{d^2 y}{d x^2}+\left[\frac{d y}{d x}\right]^2=0 ; \quad y \frac{d^2 y}{d x^x}+\left(\frac{d y}{d x}\right)^2=0$.