What is the solution for y(169-x^2)^{1/2}dy=(169+y^2)^{1/2}dx ?

The solution for y(169-x^2)^{1/2}dy=(169+y^2)^{1/2}dx is

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y(169-x^2)^{1/2}dy=(169+y^2)^{1/2}dx

y (169 - x^{2})^{\frac{1}{2}} d y = (169 + y^{2})^{\frac{1}{2}} d x

y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

Solution steps

y (169 - x^{2})^{\frac{1}{2}} d y = (169 + y^{2})^{\frac{1}{2}} d x

Solve separable ODE:

y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) + ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 338 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{13} ) + 2 c _{1} arcsin ( \frac{x}{13} ) - ( arcsin ^{4} ( \frac{x}{13} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{13} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{13} ) + 4 c _{1}^{3} arcsin ( \frac{x}{13} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

\frac{d x}{d t} = - 2 x + 1

2 y^{^{'}} + y = 0, y (0) = e - \frac{x}{2}

\frac{d y}{d x} = \frac{x ^{2} - x y - x + y}{x y - y ^{2}}

\frac{d y}{d x} = \frac{1 + y ^{2}}{x}

x^{2} (y^{2} + 1) d x + y x^{3} + 1 d y = 0

What is the solution for y(169-x^2)^{1/2}dy=(169+y^2)^{1/2}dx ?
The solution for y(169-x^2)^{1/2}dy=(169+y^2)^{1/2}dx is