What is the solution for v^2xdx+vx^2dv+vxdx+x^2dv=vxdx-xdx ?

The solution for v^2xdx+vx^2dv+vxdx+x^2dv=vxdx-xdx is x=(e^{-arctan(v)+c_{1}})/(sqrt(v^2+1))

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v^2xdx+vx^2dv+vxdx+x^2dv=vxdx-xdx

v^{2} x d x + v x^{2} d v + vx d x + x^{2} d v = vx d x - x d x

x = \frac{e ^{- a r c t a n (v) + c_{1}}}{v ^{2} + 1}

Solution steps

v^{2} x d x + v x^{2} d v + vx d x + x^{2} d v = vx d x - x d x

Solve separable ODE:

x = \frac{e ^{- a r c t a n (v) + c_{1}}}{v ^{2} + 1}

x = \frac{e ^{- a r c t a n (v) + c_{1}}}{v ^{2} + 1}

\frac{d y}{y - 4} = x^{4} d x

\frac{d y}{d x} = x^{3} (y - 2), y (0) = 4

y^{^{'}} = y cos (x), y (0) = 5

y^{^{'}} = 6 x y + 42 x - 4 y - 28

x^{2} y^{^{'}} - 6 x y + 10 y = 0

What is the solution for v^2xdx+vx^2dv+vxdx+x^2dv=vxdx-xdx ?
The solution for v^2xdx+vx^2dv+vxdx+x^2dv=vxdx-xdx is x=(e^{-arctan(v)+c_{1}})/(sqrt(v^2+1))