4.14.23 \(\left (-x^3+6 x^2 y(x)^2+1\right ) y'(x)=x \left (-4 y(x)^3+3 x y(x)+6\right )\)

ODE
\[ \left (-x^3+6 x^2 y(x)^2+1\right ) y'(x)=x \left (-4 y(x)^3+3 x y(x)+6\right ) \] ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0271848 (sec), leaf count = 424

\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} \left (x^3-1\right )}{\sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}-\frac {\sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}{6 \sqrt [3]{2} x^2}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}{12 \sqrt [3]{2} x^2}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-1\right )}{2^{2/3} \sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{108 c_1 x^4+\sqrt {\left (108 c_1 x^4-324 x^6\right ){}^2-864 x^6 \left (x^3-1\right )^3}-324 x^6}}{12 \sqrt [3]{2} x^2}\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 28

\[ \left \{ 2\,{x}^{2} \left (y \relax (x ) \right ) ^{3}-{x}^{3}y \relax (x ) -3\,{x}^{2}+{\it \_C1}+y \relax (x ) =0 \right \} \] 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[-86
4*x^6*(-1 + x^3)^3 + (-324*x^6 + 108*x^4*C[1])^2])^(1/3)) + ((1 - I*Sqrt[3])*(-3
24*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)*(-32
4*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),'implicit')

Maple raw output

2*x^2*y(x)^3-x^3*y(x)-3*x^2+_C1+y(x) = 0