The differential equation that represents all parabolas having their axis of symmetry coincident with the axis of x, is

The equation that represents a family of parabolas having their axis of symmetry coincident with the axis of x isy2=4a(x−h)This equation contains two arbitrary constants, so we shall differentiate it twice to obtain a second order differential equationDifferentiating (i) w.r.t. x, we get2ydydx=4a⇒ydydx=2aDifferentiating (ii) w.r.t. x, we getyd2ydx2+dydx2=0⇒yy2+y12=0which is the required differential equation .