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