Question

Obtain the differential equation by eliminating the arbitrary constants from the given equation, $y=c^{2}+\frac{c}{x}$

Answer 1

To obtain the differential equation by eliminating the arbitrary constants from the given equation, $y=c^{2}+\frac{c}{x}$

$y = {c}^{2} + \frac{c}{x} ...eq1 \\ \\ \frac{dy}{dx} = 0 + ( \frac{ - c}{ {x}^{2} } ) \\ \\ \frac{dy}{dx} = ( \frac{ - c}{ {x}^{2} } )...eq2 \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = - c( \frac{ - 2x}{ {x}^{4} } ) \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = ( \frac{ 2c}{ {x}^{3} } )...eq3 \\ \\ = ( - \frac{c}{ {x}^{2} } )( \frac{2}{x} ) \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} }= \frac{2}{x} \frac{dy}{dx} \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } - \frac{2}{x} \frac{dy}{dx} = 0 \\ \\ x\frac{ {d}^{2}y }{ {dx}^{2} } -2 \frac{dy}{dx} = 0$
is the required differential equation.