find the diffeential equationof the family of curves represented c(y+c)^2=x^3, where c is a parameter .
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Answer:
step-by-step explanation
Solution :
Given curve, c(y+c)
2
+x
3
=0___(1)
Differentiating w.r.t x,
2c(y+c)
dx
dy
+3x
2
=0
From eq
n
(1)
c(y+c)=
(y+c)
−x
3
∴
y+c
−2x
3
dx
dy
+3x
2
=0
⇒3=
y+c
2x
dx
dy
⇒
3
2x
dx
dy
=y+c
substituting c to equation (1)
(
3
2x
dx
dy
−y)(
3
2x
dx
dy
)
2
+x
3
=0
9
4x
2
(
dx
dy
)
2
[
3
2x
dx
dy
−y]+x
3
=0
⇒
9
4x
2
(
dx
dy
)
3
×
3
2x
−y×
9
4x
2
(
dx
dy
)
2
+x
3
=0
⇒(
3
2x
(
dx
dy
)
3
−
9
4x
2
y
(
dx
dy
))
2
+x
3
=0
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