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

Lösung

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}}}

Schritte zur Lösung

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

Solve separable ODE:

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}}}

y^{^{'}} y = 2 x + 1, y (0) = 1

ln (x) \cdot y^{^{'}} - \frac{2 y}{x} = 0

\frac{d y}{d t} = t e^{t + y}

\frac{d x}{d y} = 0.8 (28 - x)

y^{^{'}} + x y^{^{'}} - 2 y = 0