The differential equation `(dy)/(dx)=(2x)/(3y)` represents a family of hyperbolas (except when it represents a pair of lines) with eccentricity. (a)`sqrt(3/5)` (b) `sqrt(5/3)` (c)`sqrt(2/5)` (d) `sqrt(5/2)`

Options

A. `sqrt(3//5)`
B. `sqrt(5//3)`
C. `sqrt(2//5)`
D. `sqrt(5//2)`

Answer

B, D

Explanation

(2), (4)
`(dx)/(dy)=(3y)/(2x)`
`"or "int2xdx=int3ydy`
`"or "x^(2)=(3y^(2))/(2)+c`
`"or "(x^(2))/(3)-(y^(2))/(2)=(c)/(3)`
If c is positive, then
`e=sqrt(1+(2)/(3))=sqrt((5)/(3))`
If c is negative, then
`e=sqrt(1+(3)/(2))=sqrt((5)/(2))`