S1: The differential equation of parabolas having their vertices at the origin

Discussion Forum : Differential Equations

Question -

S1: The differential equation of parabolas having their vertices at the origin and foci on the x-axis is an equation whose variables are separable
S2: The differential equation of the straight lines which are at a fixed distance p from the origin is an equation of degree 2
S3: The differential equation of all conics whose both axes coincide with the axes of coordinates is an equation of order 2

Options:

A . TTT

B . TFT

C . FFT

D . TTF

Answer: Option A
:
A

S_{1}

-Equation of parabola is

y^{2} = \pm 4 a x

2 y \frac{d y}{d x} = \pm r a

D.E of parabola

\Rightarrow y^{2} = 2 y x \frac{d y}{d x}

2 \frac{d y}{y} = \frac{d x}{x}

Which is variable seperable

S_{2}

-Equation of line which is fixed distance. P from origin can be equation of tangent to circle

s^{2} + y^{2} = p^{2}

Line is

y = m x + p \sqrt{1 + m^{2}} (m = \frac{d y}{d x})

{(y - x \frac{d y}{d x})}^{2} = P (1 + {(\frac{d y}{d x})}^{2})

So, degree is 2

S_{3}

-Equation of conic whose both axis co-incide with co-ordinate axis is

a x^{2} + b y^{2} = 1

As there are two constants, so order of D.E is 2

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