ODE
\[ y'(x) \left (a x^2-b x^3+3 c x y(x)^2+2 c y(x)^3\right )+a y(x)^2+2 b x^3+3 b x^2 y(x)-c y(x)^3=0 \] ODE Classification
[_rational]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.653736 (sec), leaf count = 520
\[\left \{\left \{y(x)\to \frac {-\sqrt [3]{2} \left (\sqrt {3} \sqrt {c^3 \left (4 (a x+c_1){}^3+27 c x^2 \left (b x^2+c_1\right ){}^2\right )}+9 b c^2 x^3+9 c^2 c_1 x\right ){}^{2/3}+2 \sqrt [3]{3} a c x+2 \sqrt [3]{3} c c_1}{6^{2/3} c \sqrt [3]{\sqrt {3} \sqrt {c^3 \left (4 (a x+c_1){}^3+27 c x^2 \left (b x^2+c_1\right ){}^2\right )}+9 b c^2 x^3+9 c^2 c_1 x}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 (3 a c x+3 c c_1){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}{6 \sqrt [3]{2} c}-\frac {i \left (\sqrt {3}-i\right ) (a x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {4 (3 a c x+3 c c_1){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}\right \},\left \{y(x)\to \frac {i \left (\sqrt {3}+i\right ) (a x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {4 (3 a c x+3 c c_1){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {4 (3 a c x+3 c c_1){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}{6 \sqrt [3]{2} c}\right \}\right \}\]
Maple ✓
cpu = 0.193 (sec), leaf count = 912
\[\left [y \left (x \right ) = \frac {\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}{6 c}+\frac {-2 a x +2 \textit {\_C1}}{\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}{12 c}-\frac {-a x +\textit {\_C1}}{\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}{6 c}-\frac {2 \left (-a x +\textit {\_C1} \right )}{\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}{12 c}-\frac {-a x +\textit {\_C1}}{\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}{6 c}-\frac {2 \left (-a x +\textit {\_C1} \right )}{\left (\left (-108 b \,x^{3}+108 x \textit {\_C1} +12 \sqrt {-\frac {3 \left (-27 b^{2} c \,x^{6}+54 \textit {\_C1} b c \,x^{4}-4 a^{3} x^{3}-27 \textit {\_C1}^{2} c \,x^{2}+12 \textit {\_C1} \,a^{2} x^{2}-12 \textit {\_C1}^{2} a x +4 \textit {\_C1}^{3}\right )}{c}}\right ) c^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[2*b*x^3 + 3*b*x^2*y[x] + a*y[x]^2 - c*y[x]^3 + (a*x^2 - b*x^3 + 3*c*x*y[x]^2 + 2*c*y[x]^3)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2*3^(1/3)*a*c*x + 2*3^(1/3)*c*C[1] - 2^(1/3)*(9*b*c^2*x^3 + 9*c^2*x*C[1] + Sqrt[3]*Sqrt[c^3*(4*(a*x + C[1])^3 + 27*c*x^2*(b*x^2 + C[1])^2)])^(2/3))/(6^(2/3)*c*(9*b*c^2*x^3 + 9*c^2*x*C[1] + Sqrt[3]*Sqrt[c^3*(4*(a*x + C[1])^3 + 27*c*x^2*(b*x^2 + C[1])^2)])^(1/3))}, {y[x] -> ((-I)*(-I + Sqrt[3])*(a*x + C[1]))/(2^(2/3)*(27*b*c^2*x^3 + 27*c^2*x*C[1] + Sqrt[4*(3*a*c*x + 3*c*C[1])^3 + (27*b*c^2*x^3 + 27*c^2*x*C[1])^2])^(1/3)) + ((1 - I*Sqrt[3])*(27*b*c^2*x^3 + 27*c^2*x*C[1] + Sqrt[4*(3*a*c*x + 3*c*C[1])^3 + (27*b*c^2*x^3 + 27*c^2*x*C[1])^2])^(1/3))/(6*2^(1/3)*c)}, {y[x] -> (I*(I + Sqrt[3])*(a*x + C[1]))/(2^(2/3)*(27*b*c^2*x^3 + 27*c^2*x*C[1] + Sqrt[4*(3*a*c*x + 3*c*C[1])^3 + (27*b*c^2*x^3 + 27*c^2*x*C[1])^2])^(1/3)) + ((1 + I*Sqrt[3])*(27*b*c^2*x^3 + 27*c^2*x*C[1] + Sqrt[4*(3*a*c*x + 3*c*C[1])^3 + (27*b*c^2*x^3 + 27*c^2*x*C[1])^2])^(1/3))/(6*2^(1/3)*c)}}
Maple raw input
dsolve((a*x^2-b*x^3+3*c*x*y(x)^2+2*c*y(x)^3)*diff(y(x),x)+2*b*x^3+3*b*x^2*y(x)+a*y(x)^2-c*y(x)^3 = 0, y(x))
Maple raw output
[y(x) = 1/6/c*((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)+2*(-a*x+_C1)/((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3), y(x) = -1/12/c*((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)-(-a*x+_C1)/((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)-1/2*I*3^(1/2)*(1/6/c*((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)-2*(-a*x+_C1)/((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)), y(x) = -1/12/c*((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)-(-a*x+_C1)/((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)+1/2*I*3^(1/2)*(1/6/c*((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3)-2*(-a*x+_C1)/((-108*b*x^3+108*x*_C1+12*(-3*(-27*b^2*c*x^6+54*_C1*b*c*x^4-4*a^3*x^3-27*_C1^2*c*x^2+12*_C1*a^2*x^2-12*_C1^2*a*x+4*_C1^3)/c)^(1/2))*c^2)^(1/3))]