Internal problem ID [3371]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 23
Problem number: 631.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (1-3 y x^{2}+6 y^{2}\right ) y^{\prime }+x^{2}-3 x y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 797
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), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}-\frac {12 \left (\frac {1}{6}-\frac {x^{4}}{16}\right )}{\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y \relax (x ) = -\frac {\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-108 x^{2}-144 x^{3}-432 c_{1}+27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 x^{3} c_{1}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.234 (sec). Leaf size: 570
DSolve[(1-3 x^2 y[x]+6 y[x]^2)y'[x]+x^2-3 x y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^2}{4}-\frac {\sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {6-\frac {9 x^4}{4}}{3\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}} \\ y(x)\to \frac {x^2}{4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}}{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]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}} \\ y(x)\to \frac {x^2}{4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}}{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]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (\frac {9}{4} \left (-3 x^4+16 x+12\right ) x^2+108 c_1\right ){}^2}+108 c_1}} \\ \end{align*}