Give the correct answer and solve the given equation [x-y arctan(frac{y}{x})]dx+x arctan (frac{y}{x})dy=0

FizeauV 2021-03-06 Answered

Give the correct answer and solve the given equation [xyarctan(yx)]dx+xarctan(yx)dy=0

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Expert Answer

Arnold Odonnell
Answered 2021-03-07 Author has 109 answers

We will first write this equation as
xarctan(yx)dydx=yarctan(yx)x (1)
Notice that arctan(yx)=0
means that yx=0, so y=0. However, this is
not a solution of the given equation.
Thus, arctan(yx)0, so we can divide the equation (1)
by xarctan(yx)
dydx=yx1arctan(yx) (2)
Make the substitution u=yx or y=ux. Then
dydx=ddx(ux)=xdydx+u (3)
(use the Chain Rule). Furthermore, (2) becomes
dydx=u1arctan(u) (4)
Combining (3) and (4), we get xdudx+u=u1arctan(u)xdudx=1arctan(u)
Notice that this is a separable equation! It can be written as
arctan(u)du=1xdx
Integrate both sides:
arctan(u)du=dxd (5)
Clearly,
dxd=ln|x|+C2,
where C2 is some constant.
For the other integral, we use the integration by parts:
arctan(u)du={(t=arctan(u)dv=du),(dt=du1+u2v=u)}
=uarctan(u)udu1+u2du
=uarctan(u)12ln|1+u2|+C1
=uarctan(u)ln1+u2+C1
where C1 is some constant.
(In last equality we need used that 1+u2>0, so |1+u2|=1+u2,
and the property of the logarithmic function: xlny=lnyx.)
Finnaly. (5) becomes
=uarctan(u)ln1+u2=ln|x|+C,
where C=C2C1

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