ODE
\[ x^2 (a x+3 y(x))+\left (x^3+y(x)^3\right ) y'(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.515751 (sec), leaf count = 1430
\[\left \{\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}-\sqrt {-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}+\sqrt {-\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}+\sqrt {\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {6 \sqrt {2} x^3}{\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}-\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}+\frac {\sqrt [3]{3} \left (e^{4 c_1}-a x^4\right )}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}-\sqrt {\frac {\sqrt [3]{3} a x^4-\sqrt [3]{3} e^{4 c_1}+\left (9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}\right ){}^{2/3}}{\sqrt [3]{9 x^6+\sqrt {3} \sqrt {27 x^{12}+\left (e^{4 c_1}-a x^4\right ){}^3}}}}}{\sqrt {2} \sqrt [3]{3}}\right \}\right \}\]
Maple ✓
cpu = 0.232 (sec), leaf count = 29
\[\left [y \left (x \right ) = \frac {\RootOf \left (a \,x^{4} \textit {\_C1}^{\frac {4}{3}}+4 x^{3} \textit {\_C1} \textit {\_Z} +\textit {\_Z}^{4}-1\right )}{\textit {\_C1}^{\frac {1}{3}}}\right ]\] Mathematica raw input
DSolve[x^2*(a*x + 3*y[x]) + (x^3 + y[x]^3)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] - Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^(1/3))}, {y[x] -> (Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^(1/3))}, {y[x] -> -((Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) + (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^(1/3)))}, {y[x] -> (-Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)] + Sqrt[(3^(1/3)*(E^(4*C[1]) - a*x^4))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) - (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3) + (6*Sqrt[2]*x^3)/Sqrt[(-(3^(1/3)*E^(4*C[1])) + 3^(1/3)*a*x^4 + (9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(2/3))/(9*x^6 + Sqrt[3]*Sqrt[27*x^12 + (E^(4*C[1]) - a*x^4)^3])^(1/3)]])/(Sqrt[2]*3^(1/3))}}
Maple raw input
dsolve((x^3+y(x)^3)*diff(y(x),x)+x^2*(a*x+3*y(x)) = 0, y(x))
Maple raw output
[y(x) = RootOf(a*x^4*_C1^(4/3)+4*x^3*_C1*_Z+_Z^4-1)/_C1^(1/3)]