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

解

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}

解答ステップ

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

分離可能 ODE を解く:

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}