xy^3dx+e^{x^3}dy=0

解答

x y^{3} d x + e^{x^{3}} d y = 0

y = \frac{- 0 + 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}, y = \frac{- 0 - 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}

求解步骤

x y^{3} d x + e^{x^{3}} d y = 0

求解可分离方程:

y = \frac{- 0 + 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}, y = \frac{- 0 - 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}

y = \frac{- 0 + 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}, y = \frac{- 0 - 0 ^{2} - 4 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1} \cdot 1}{2 \frac{1}{2} \cdot \frac{2}{3} \cdot 3 \cdot \frac{1}{2} - \frac{( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} Γ ( \frac{1}{\frac{3}{2}} , - ( ( 3 - ( x ^{2} ) ^{\frac{3}{2}} ) ^{2} ) ^{\frac{3}{2}} )}{\frac{3}{2} \frac{3}{2} - ( ( 3 - ( x ^{2} ) ) ^{2} ) ^{\frac{3}{2}}} \cdot 2 + 2 c _{1}}

\frac{d y}{d x} = \frac{y sin ( x )}{y + 1}

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

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

d y = 3 x^{2} (4 - y) d x

y - y^{^{'}} - 6 y = 18