Question

xdx+ydy=a²(xdy-ydx)÷x²+y²​

Answer 1

Answer:

think that your question may be

xdx+ydy={a}^2(xdy+ydx)/({x}^2+{y}^2)xdx+ydy=a

2

(xdy+ydx)/(x

2

+y

2

)

({x}^2+{y}^2)(xdx+ydy)={a}^2(xdy+ydx)..(1)(x

2

+y

2

)(xdx+ydy)=a

2

(xdy+ydx)..(1)

\begin{gathered}Take, \\u={x}^2+{y}^2\\\frac{du}{dx}=2x+2y\frac{dy}{dx}\\\frac{du}{dx}=2(x+y\frac{dy}{dx})\\\frac{du}{2}=xdx+ydy\end{gathered}

Take,

u=x

2

+y

2

dx

du

=2x+2y

dx

dy

dx

du

=2(x+y

dx

dy

)

2

du

=xdx+ydy

\begin{gathered}v=xy\\\frac{dv}{dx}=x\frac{dy}{dx}+y.1\\dv=xdy+ydx\\Now (1) becomes\\\end{gathered}

v=xy

dx

dv

=x

dx

dy

+y.1

dv=xdy+ydx

Now(1)becomes

\begin{gathered}u\frac{du}{2}={a}^2dv\\Integrating\\\frac{1}{2}\int\:u\:du={a}^2\int\:dv\\\end{gathered}

u

2

du

=a

2

dv

Integrating

2

1

∫udu=a

2

∫dv

\begin{gathered}\frac{{u}^2}{2}={a}^2v+c\\\frac{{({x}^2+{y}^2)}^2}{2}={a}^2xy+c\end{gathered}

2

u

2

=a

2

v+c

2

(x

2

+y

2

)

2

=a

2

xy+c

Step-by-step explanation:

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