y(25-x^2)^{1/2}dy=(25+y^2)^{1/2}dx

해법

y (25 - x^{2})^{\frac{1}{2}} d y = (25 + y^{2})^{\frac{1}{2}} d x

y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

솔루션 단계

y (25 - x^{2})^{\frac{1}{2}} d y = (25 + y^{2})^{\frac{1}{2}} d x

분리 가능한 ODE 해결:

y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}

y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) + ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}, y = - \frac{- 50 + c _{1}^{2} + arcsin ^{2} ( \frac{x}{5} ) + 2 c _{1} arcsin ( \frac{x}{5} ) - ( arcsin ^{4} ( \frac{x}{5} ) + 4 c _{1} arcsin ^{3} ( \frac{x}{5} ) + 6 c _{1}^{2} arcsin ^{2} ( \frac{x}{5} ) + 4 c _{1}^{3} arcsin ( \frac{x}{5} ) + c _{1}^{4} ) ^{\frac{1}{2}}}{2}