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Question
Find the differential equation of the family of a parabola with foci at the origin and axis along the x-axis
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Solution
Equation of parabola with foci at the origin and axis along the x-axis is
y2 = 4a(x + a) .......(1)
Differentiate w.r.t. x
`2y ("d"y)/("d"x)` = 4a(1 + 0)
2y = `("d"y)/("d"x)` = 4a
⇒ a = `y/2, ("d"y)/("d"x)`
Substitute the value of a = `y/2 ("d"y)/("d"x)` in equation (1)
y2 = `4 (y/2 ("d"y)/("d"x)) [x + y/2 ("d"y)/("d"x)]`
y2 = `2y ("d"y)/("d"x) (x + y/2 ("d"y)/("d"x))`
÷ By y on both sides
y = `2("d"y)/("d"x) (x + y/2 ("d"y)/("d"x))`
y = `2 ("d"y)/("d"x) ((2x + y ("d"y)/("d"x))/2)`
⇒ y = `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2`
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