The differential equation of all parabolas whose axis are parallel to the y-axis is

Question

The differential equation of all parabolas whose axis are parallel to the y-axis is (A) (d3y/dx3)=0 (B) (d2x/dy2)=C (C) (d3y/dx3)+(d2x/dy2)=0 (D) (d2y/dx2)+2 (dy/dx)=C

Tardigrade Admin · Accepted Answer

The equation of a member of the family of parabolas having axis parallel to

y

-axis is

y = A x^{2} + B x + C ... (1)

where A, B, and C are arbitrary constants
Differentiating equation (1) w.r.t.

x

, we get

\frac{d y}{d x} = 2 A x + B ... (2)

which on again differentiating w.r.t. x gives

\frac{d ^{2} y}{d x ^{2}} = 2 A ... (3)

Differentiating (3) w.r.t. x, we get

\frac{d ^{3} y}{d x ^{3}} = 0

Q. The differential equation of all parabolas whose axis are parallel to the y-axis is

$\frac{d ^{3} y}{d x ^{3}} = 0$

$\frac{d ^{2} x}{d y ^{2}} = C$

$\frac{d ^{3} y}{d x ^{3}} + \frac{d ^{2} x}{d y ^{2}} = 0$

$\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} = C$

Q. The differential equation of all parabolas whose axis are parallel to the y-axis is

dx3d3y​=0

dy2d2x​=C

dx3d3y​+dy2d2x​=0

dx2d2y​+2dxdy​=C

$\frac{d ^{3} y}{d x ^{3}} = 0$

$\frac{d ^{2} x}{d y ^{2}} = C$

$\frac{d ^{3} y}{d x ^{3}} + \frac{d ^{2} x}{d y ^{2}} = 0$

$\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} = C$