Single Choice

The differential equation of all parabolas whose axis is y-axis is:

A$$x\dfrac {d^{2}y}{dx^{2}} - \dfrac {dy}{dx} = 0$$
Correct Answer
B$$x\dfrac {d^{2}y}{dx^{2}} + \dfrac {dy}{dx} = 0$$
C$$\dfrac {d^{2}y}{dx^{2}} - y = 0$$
D$$\dfrac {d^{2}y}{dx^{2}} - \dfrac {dy}{dx} = 0$$

Solution

$$axis = y\ axis$$
vertex is $$(0, k)$$
Equation of parabola is
$$(x - 0)^{2} = 4a(y - k)$$
$$x^{2} = 4ay - 4ak$$
Differentiate w.r.t $$x$$
$$2x = 4a \dfrac {dy}{dx}$$
$$x = 2a \dfrac {dy}{dx}$$
$$\therefore \dfrac {1}{2a} = \dfrac {1}{x} \dfrac {dy}{dx}$$

Differentiate w.r.t $$x$$,
$$\dfrac {d}{dx} \left (\dfrac {1}{x} . \dfrac {dy}{dx}\right ) = \dfrac {d}{dx} \left (\dfrac {1}{2a}\right )$$
$$\dfrac {1}{x} . \dfrac {d^{2}y}{dx^{2}} + \dfrac {dy}{dx} \left (-\dfrac {1}{x^{2}}\right ) = 0$$
$$\therefore x \dfrac {d^{2}y}{dx^{2}} - \dfrac {dy}{dx} = 0$$

SIMILAR QUESTIONS

Differential Equations

Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If k is a positive constant then differential equation involving the rate of change of the radius of the rain drop is

Differential Equations

The differential equation of all parabolas whose axes are parallel to $$\mathrm{y}$$ -axis is:

Differential Equations

$$\displaystyle \left ( x^{3}-y^{3} \right )dx+xy^{2}dy= 0.$$ Solving this we get $$\displaystyle \frac{k}{x}=e^{y^{m}/nx^{r}} $$.Find $$m+n+r$$ ?

Differential Equations

If $$y=e^{m sin^{-1}x}$$, then show that $$(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}-m^2y=0$$.

Differential Equations

If $$\displaystyle 2x=y^{\tfrac{1}{5}}+y^{-\tfrac{1}{5}}$$ and $$\displaystyle (x^2-1)\dfrac{d^2y}{dx^2}+\lambda x\dfrac{dy}{dx}+ky=0$$, then $$\lambda +k$$ is equal to:

Differential Equations

Let $$(x,y,z)$$ be points with integer coordinates satisfying the system of homogeneous equations: $$3x-y-z=0$$ $$-3x+z=0$$ $$-3x+2y+z=0$$. Then the number of such points which lie inside a sphere of radius $$10$$ centered at the origin is

Differential Equations

The differential equation of the family of circles touching y-axis at the origin is:

Differential Equations

If $$ y={ \left( \tan ^{ -1 }{ x } \right) }^{ 2 }$$, then $${ \left( { x }^{ 2 }+1 \right) }^{ 2 }\cfrac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +2x\left( { x }^{ 2 }+1 \right) \cfrac { dy }{ dx } =$$

Differential Equations

if $$y = A{e^{ - kt}}\cos (pt + c)$$ , then prove that $${{{d^2}y} \over {d{t^2}}} + 2k{{dy} \over {dt}} + {n^2}y = 0$$ , where $${n^2} = {p^2} + {k^2}$$

Differential Equations

Find the differential equation of all the parabola having axis parallel to the $$x-$$axis.

Contact Details