y^'=0.5(ye^x)^{1/3}

解

y^{^{'}} = 0.5 (y e^{x})^{\frac{1}{3}}

y = \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}, y = - \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}

解答ステップ

y^{'} (x) = 0.5 (y e^{x})^{\frac{1}{3}}

分離可能 ODE を解く:

y = \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}, y = - \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}

y = \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}, y = - \frac{( 27 e ^{x} + 54 c _{1} e ^{\frac{2 x}{3}} + 36 c _{1}^{2} e ^{\frac{x}{3}} + 8 c _{1}^{3} ) ^{\frac{1}{2}}}{3 \cdot 3 ^{\frac{1}{2}}}