4.13.34 \(\left (-3 x^2 y(x)+6 y(x)^2+1\right ) y'(x)+x^2-3 x y(x)^2=0\)

ODE
\[ \left (-3 x^2 y(x)+6 y(x)^2+1\right ) y'(x)+x^2-3 x y(x)^2=0 \] ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.0259497 (sec), leaf count = 570

\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}{6 \sqrt [3]{2}}+\frac {6-\frac {9 x^4}{4}}{3\ 2^{2/3} \sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}+\frac {x^2}{4}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}{12 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}+\frac {x^2}{4}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}{12 \sqrt [3]{2}}-\frac {\left (1-i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{\sqrt {\left (108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2\right ){}^2+4 \left (6-\frac {9 x^4}{4}\right )^3}+108 c_1-\frac {27 x^6}{4}+36 x^3+27 x^2}}+\frac {x^2}{4}\right \}\right \}\]

Maple
cpu = 0.018 (sec), leaf count = 27

\[ \left \{ -{\frac {3\,{x}^{2} \left (y \relax (x ) \right ) ^{2}}{2}}+{\frac {{x}^{3}}{3}}+2\, \left (y \relax (x ) \right ) ^{3}+y \relax (x ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x^2 - 3*x*y[x]^2 + (1 - 3*x^2*y[x] + 6*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> x^2/4 + (6 - (9*x^4)/4)/(3*2^(2/3)*(27*x^2 + 36*x^3 - (27*x^6)/4 + 108
*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])
^(1/3)) - (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + 
(27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])^(1/3)/(6*2^(1/3))}, {y[x] -> x^2/4
 - ((1 + I*Sqrt[3])*(6 - (9*x^4)/4))/(6*2^(2/3)*(27*x^2 + 36*x^3 - (27*x^6)/4 + 
108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^
2])^(1/3)) + ((1 - I*Sqrt[3])*(27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*
(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[1])^2])^(1/3))/(12*2^(
1/3))}, {y[x] -> x^2/4 - ((1 - I*Sqrt[3])*(6 - (9*x^4)/4))/(6*2^(2/3)*(27*x^2 + 
36*x^3 - (27*x^6)/4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (
27*x^6)/4 + 108*C[1])^2])^(1/3)) + ((1 + I*Sqrt[3])*(27*x^2 + 36*x^3 - (27*x^6)/
4 + 108*C[1] + Sqrt[4*(6 - (9*x^4)/4)^3 + (27*x^2 + 36*x^3 - (27*x^6)/4 + 108*C[
1])^2])^(1/3))/(12*2^(1/3))}}

Maple raw input

dsolve((1-3*x^2*y(x)+6*y(x)^2)*diff(y(x),x)+x^2-3*x*y(x)^2 = 0, y(x),'implicit')

Maple raw output

-3/2*x^2*y(x)^2+1/3*x^3+2*y(x)^3+y(x)+_C1 = 0