xdx+ydy=a²(xdy-ydx)÷x²+y²
Answers
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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