ODE
\[ \left (-x^3+6 x^2 y(x)^2+1\right ) y'(x)=x \left (-4 y(x)^3+3 x y(x)+6\right ) \] ODE Classification
[_exact, _rational]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.559445 (sec), leaf count = 424
\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}-\frac {\sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{6 \sqrt [3]{2} x^2}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-324 x^6+108 c_1 x^4+\sqrt {-864 x^6 \left (x^3-1\right )^3+\left (-324 x^6+108 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}\right \}\right \}\]
Maple ✓
cpu = 0.038 (sec), leaf count = 601
\[\left [y \left (x \right ) = \frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}{6 x}+\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}-1}{2 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}{6 x}-\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}-1}{2 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}{6 x}-\frac {x^{3}-1}{x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 \textit {\_C1} \,x^{4}+27 \textit {\_C1}^{2} x^{2}-6 x^{3}+2}-54 x \textit {\_C1} \right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[(1 - x^3 + 6*x^2*y[x]^2)*y'[x] == x*(6 + 3*x*y[x] - 4*y[x]^3),y[x],x]
Mathematica raw output
{{y[x] -> -((2^(1/3)*(-1 + x^3))/(-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3)) - (-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3)/(6*2^(1/3)*x^2)}, {y[x] -> ((1 + I*Sqrt[3])*(-1 + x^3))/(2^(2/3)*(-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3)) + ((1 - I*Sqrt[3])*(-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3))/(12*2^(1/3)*x^2)}, {y[x] -> ((1 - I*Sqrt[3])*(-1 + x^3))/(2^(2/3)*(-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3)) + ((1 + I*Sqrt[3])*(-324*x^6 + 108*x^4*C[1] + Sqrt[-864*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3))/(12*2^(1/3)*x^2)}}
Maple raw input
dsolve((1-x^3+6*x^2*y(x)^2)*diff(y(x),x) = (6+3*x*y(x)-4*y(x)^3)*x, y(x))
Maple raw output
[y(x) = 1/6/x*(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)+(x^3-1)/x/(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3), y(x) = -1/12/x*(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)-1/2*(x^3-1)/x/(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)-1/2*I*3^(1/2)*(1/6/x*(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)-(x^3-1)/x/(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)), y(x) = -1/12/x*(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)-1/2*(x^3-1)/x/(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)+1/2*I*3^(1/2)*(1/6/x*(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3)-(x^3-1)/x/(162*x^3+6*3^(1/2)*(-2*x^9+249*x^6-162*_C1*x^4+27*_C1^2*x^2-6*x^3+2)^(1/2)-54*x*_C1)^(1/3))]