Question

Find differential equation of all parabolas x^2=4a(y+k) .

Answer 1

Question:-

Find differential equation of all parabolas

x² = 4a(y + k).

Given:-

• Equation of parabolas x² = 4a(y + k).

To Find:-

• Differential equation.

Solution:-

Equation of parabolas is x² = 4a(y + k).

Differentiate w.r.t.x we get,

$\sf \large \Rightarrow2x = 4a \: \frac{dy}{dx}$

$\sf \large x = 2a \: \frac{dy}{dx} \\ \sf \large\therefore \frac{1}{2a} = \frac{1}{x} \: \frac{dy}{dx}$

Again differentiate w.r.t.x we get,

$\sf \large \frac{d}{dx} ( \frac{1}{x} . \frac{dy}{dx} ) = \frac{d}{dx} ( \frac{1}{2a} )$

$\sf \large \Rightarrow \frac{1}{x} . \frac{d {}^{2}y }{d x{}^{2} } + \frac{dy}{dx} .( - \frac{1}{x {}^{2} } ) = 0$

$\sf \large \Rightarrow x \frac{d {}^{2}y }{dx {}^{2} } - \frac{dy}{dx} = 0$

Answer:-

${ \boxed{ \sf \large \pink{Hence, \: Differentiate \: equation \: is \: x \frac{d {}^{2}y }{dx {}^{2} } = 0. } }}$