ODE
\[ 2 x^3+3 x^2 y(x)+\left (-x^3+x^2+3 x y(x)^2+2 y(x)^3\right ) y'(x)-y(x)^3+y(x)^2=0 \] ODE Classification
[_rational]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.392026 (sec), leaf count = 361
\[\left \{\left \{y(x)\to \frac {\sqrt [3]{2} (x+c_1)}{\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}-\frac {\sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}{3 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}-6 i \sqrt [3]{2} \left (\sqrt {3}-i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}\right \},\left \{y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x\right ){}^{2/3}+6 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) (x+c_1)}{12 \sqrt [3]{27 x^3+\sqrt {729 \left (x^3+c_1 x\right ){}^2+108 (x+c_1){}^3}+27 c_1 x}}\right \}\right \}\]
Maple ✓
cpu = 0.178 (sec), leaf count = 662
\[\left [y \left (x \right ) = \frac {\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}+\frac {\textit {\_C1}}{6}\right )}{\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}{12}+\frac {x +\frac {\textit {\_C1}}{2}}{\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 x +\textit {\_C1}}{\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}{12}+\frac {x +\frac {\textit {\_C1}}{2}}{\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 x +\textit {\_C1}}{\left (-108 x^{3}-54 x \textit {\_C1} +6 \sqrt {324 x^{6}+324 \textit {\_C1} \,x^{4}+81 \textit {\_C1}^{2} x^{2}+6 \textit {\_C1}^{3}+36 x \,\textit {\_C1}^{2}+72 x^{2} \textit {\_C1} +48 x^{3}}\right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[2*x^3 + 3*x^2*y[x] + y[x]^2 - y[x]^3 + (x^2 - x^3 + 3*x*y[x]^2 + 2*y[x]^3)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (2^(1/3)*(x + C[1]))/(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(1/3) - (27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(1/3)/(3*2^(1/3))}, {y[x] -> ((-6*I)*2^(1/3)*(-I + Sqrt[3])*(x + C[1]) + 2^(2/3)*(1 - I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(1/3))}, {y[x] -> ((6*I)*2^(1/3)*(I + Sqrt[3])*(x + C[1]) + 2^(2/3)*(1 + I*Sqrt[3])*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(2/3))/(12*(27*x^3 + 27*x*C[1] + Sqrt[108*(x + C[1])^3 + 729*(x^3 + x*C[1])^2])^(1/3))}}
Maple raw input
dsolve((x^2-x^3+3*x*y(x)^2+2*y(x)^3)*diff(y(x),x)+2*x^3+3*x^2*y(x)+y(x)^2-y(x)^3 = 0, y(x))
Maple raw output
[y(x) = 1/6*(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)-6*(1/3*x+1/6*_C1)/(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3), y(x) = -1/12*(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)+3*(1/3*x+1/6*_C1)/(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/6*(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)+6*(1/3*x+1/6*_C1)/(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)), y(x) = -1/12*(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)+3*(1/3*x+1/6*_C1)/(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/6*(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3)+6*(1/3*x+1/6*_C1)/(-108*x^3-54*x*_C1+6*(324*x^6+324*_C1*x^4+81*_C1^2*x^2+6*_C1^3+36*_C1^2*x+72*_C1*x^2+48*x^3)^(1/2))^(1/3))]