If x+y=m, x^3+y^3=n, what is the value of x^2+y^2 if m is not 0?
Answers
Answer:
If x+y=m, x^3+y^3=n, what is the value of x^2+y^2 if m is not 0?
Answer:
ANSWER
Given: x
m
y
n
=(x+y)
m+n
Take log on both side
logx
m
+logy
n
+(m+n)log=y(x+y)
mlogx+nlogy=(m+n)log(x+y)
Differentiating w.r.t x on both sides we get,
m.
x
1
+n
y
1
dx
dy
=(m+n).
(x+y)
1
(1+
dx
dy
)
x
m
+
y
n
dx
dy
=
x+y
m+n
(1+
dx
dy
)
dx
dy
(
y
n
−
x+y
m+n
)=
x+y
m+n
−
x
m
dx
dy
(
y(x+y)
n(x+y)−y(m+n)
)=
x(x+y)
x(m+n)−m(x+y)
On simplifying we get,
dx
dy
(
y
nx−my
)=
x
nx−my
∴
dx
dy
=
x
y
Differentiating on both sides w.r.t x we get,
Apply quotient rule
dx
2
d
2
y
=
x
2
x.
dx
dy
−y.1
⇒
x
2
x.
dx
dy
−y
This can be written as
x
1
dx
dy
−
x
2
y
But
dx
dy
=
x
y
So substituting this we get,
x
2
y
−
x
2
y
=0
Hence proved
Step-by-step explanation:
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