What is the solution for y(81-x^2)^{1/2}dy=(81+y^2)^{1/2}dx ?

The solution for y(81-x^2)^{1/2}dy=(81+y^2)^{1/2}dx is

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y(81-x^2)^{1/2}dy=(81+y^2)^{1/2}dx

y (81 - x^{2})^{\frac{1}{2}} d y = (81 + y^{2})^{\frac{1}{2}} d x

y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

Solution steps

y (81 - x^{2})^{\frac{1}{2}} d y = (81 + y^{2})^{\frac{1}{2}} d x

Solve separable ODE:

y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) + ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 162 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{9} ) + 2 c _{1} arcsin ( \frac{x}{9} ) - ( arcsin ^{4} ( \frac{x}{9} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{9} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{9} ) + 4 c _{1}^{3} arcsin ( \frac{x}{9} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

What is the solution for y(81-x^2)^{1/2}dy=(81+y^2)^{1/2}dx ?
The solution for y(81-x^2)^{1/2}dy=(81+y^2)^{1/2}dx is