Question

Form the differential equation representing the family of curves `y² = m(a² - x²) by eliminating the arbitrary constants 'm' and 'a'. 

Answer 1

The equation y² = m(a² - x²) where m and a are arbitrary constants.

y² = m(a² - x²)   ......(i)

Differentiate (i) w.r.t.x.

`2"y"(d"y")/(d"x")` = -2mx  ...(ii)

⇒ -2m = `2 ("y")/("x") (d"y")/(d"x")`

Differentiate (ii) w.r.t.x.

`2["y" (d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2] ` = -2m  .....(iii)

From (ii) and (iii), we get

`2["y" (d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2] = 2 ("y")/("x") (d"y")/(d"x")`

`"y"(d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2-("y"/"x")(d"y")/(d"x")` = 0

therefore the required differential equation is `"y"(d^2"y")/(d"x"^2) + ((d"y")/(d"x"))^2-("y"/"x")(d"y")/(d"x")` = 0