Internal problem ID [15142]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 12. Miscellaneous problems. Exercises page 93
Problem number: 280.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {3 y^{2} x +\left (3 y x^{2}-6 y^{2}-1\right ) y^{\prime }=x^{2}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 771
dsolve((3*x*y(x)^2-x^2)+(3*x^2*y(x)-6*y(x)^2-1)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y &= \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 c_{1} x^{3}-27 x^{4}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {3 x^{4}-8}{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 c_{1} x^{3}-27 x^{4}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y &= \frac {24+i \left (-24+9 x^{4}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ y &= \frac {24+i \left (-9 x^{4}+\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}+24\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}+432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}-864 c_{1} x^{3}-648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.603 (sec). Leaf size: 570
DSolve[(3*x*y[x]^2-x^2)+(3*x^2*y[x]-6*y[x]^2-1)*y'[x]==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 {27 x^6}{4}+36 x^3+27 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 {27 x^6}{4}+36 x^3+27 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 {27 x^6}{4}+36 x^3+27 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 {27 x^6}{4}+36 x^3+27 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 {27 x^6}{4}+36 x^3+27 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 {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\ \end{align*}