4.14.43 \(y'(x) \left (a x^2-b x^3+3 c x y(x)^2+2 c y(x)^3\right )+a y(x)^2+2 b x^3+3 b x^2 y(x)-c y(x)^3=0\)

ODE
\[ y'(x) \left (a x^2-b x^3+3 c x y(x)^2+2 c y(x)^3\right )+a y(x)^2+2 b x^3+3 b x^2 y(x)-c y(x)^3=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0790246 (sec), leaf count = 524

\[\left \{\left \{y(x)\to \frac {-\sqrt [3]{2} \left (\sqrt {3} \sqrt {c^3 \left (4 \left (a x+c_1\right ){}^3+27 c x^2 \left (b x^2+c_1\right ){}^2\right )}+9 b c^2 x^3+9 c^2 c_1 x\right ){}^{2/3}+2 \sqrt [3]{3} a c x+2 \sqrt [3]{3} c c_1}{6^{2/3} c \sqrt [3]{\sqrt {3} \sqrt {c^3 \left (4 \left (a x+c_1\right ){}^3+27 c x^2 \left (b x^2+c_1\right ){}^2\right )}+9 b c^2 x^3+9 c^2 c_1 x}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {4 \left (3 a c x+3 c c_1\right ){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x\right ){}^{2/3}-6 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) c \left (a x+c_1\right )}{12 c \sqrt [3]{\sqrt {4 \left (3 a c x+3 c c_1\right ){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {4 \left (3 a c x+3 c c_1\right ){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x\right ){}^{2/3}+6 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) c \left (a x+c_1\right )}{12 c \sqrt [3]{\sqrt {4 \left (3 a c x+3 c c_1\right ){}^3+\left (27 b c^2 x^3+27 c^2 c_1 x\right ){}^2}+27 b c^2 x^3+27 c^2 c_1 x}}\right \}\right \}\]

Maple
cpu = 0.199 (sec), leaf count = 36

\[ \left \{ {\frac {-c \left (y \relax (x ) \right ) ^{3}+ \left (-ax+{\it \_C1} \right ) y \relax (x ) -b{x}^{3}+{\it \_C1}\,x}{x+y \relax (x ) }}=0 \right \} \] 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] -> ((-6*I)*2^(1/3)*(-I + Sqrt[3])*c*(a*
x + C[1]) + 2^(2/3)*(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])^(2/3))/(12*c*(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))}, {y[x] -> ((6*I)*2^(1/3)*(I + Sqrt[3])*c*(a*x + C[1]) + 2^(2/3)*(
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])^(2/3))/(12*c*(27*b*c^2*x^3 + 27*c^2*x*C[1] + S
qrt[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))}}

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),'implicit')

Maple raw output

(-c*y(x)^3+(-a*x+_C1)*y(x)-b*x^3+_C1*x)/(x+y(x)) = 0