Give the correct answer and solve the given equation displaystyle{left({x}+{y}right)}{left.{d}{x}right.}+{left({x}-{y}right)}{left.{d}{y}right.}={0}

avissidep 2021-01-05 Answered
Give the correct answer and solve the given equation
(x+y)dx+(xy)dy=0
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Expert Answer

SchulzD
Answered 2021-01-06 Author has 83 answers
First rewrite it as (xy)dy=(x+y)dx (1)
Suppose that xy=0, so y=x. Then
0=2x,
so y=x is not a solution of the starding equation. Therefore, yx, and we can divide (1) by xy:
dy=(x+y)xydx=x+yyxdx
We can also write this as
y=dydx=x+yyx=yx+2xyx=1+2xyx=1+2yx1 (2)
Now use the substitution
u=yx,
which we can also write as
yux,
to get that
y=ux+u (3)
From(2),
y=1+2yx1=1+2u1
Combining this and (3) we get
ux+u=1+2u1
Furthermore,
ux=1u+2u1=(1u)(u1)+2u1=(u1)2+2u1=(u1)2+2u1
Now write u=dudx:
dudxx=(u1)2+2u1
Whis is a separable equation, because we can write it in the form h
h(u)du=g(x)dx
To get this, first divide the equation by (u1)2+2u1 (notice that this is always nonzero!):
u1(u1)2+2dudxx=1
Now divide the equation by x:
u1(u1)2+2dudx=1x
Finally, "multiply" thq equation by dx:

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