Internal problem ID [7868]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 288.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (6 y^{2}-3 y x^{2}+1\right ) y^{\prime }-3 x y^{2}+x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 587
dsolve((6*y(x)^2-3*x^2*y(x)+1)*diff(y(x),x)-3*x*y(x)^2+x=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 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 (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y \relax (x ) = -\frac {\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}-\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{24}+\frac {1-\frac {3 x^{4}}{8}}{\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4}+\frac {i \sqrt {3}\, \left (\frac {\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {2-\frac {3 x^{4}}{4}}{\left (-324 x^{2}-432 c_{1}+27 x^{6}+12 \sqrt {-81 x^{8}-162 x^{6} c_{1}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.836 (sec). Leaf size: 538
DSolve[(6*y[x]^2-3*x^2*y[x]+1)*y'[x]-3*x*y[x]^2+x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{36} \left (9 x^2-3 \sqrt [3]{3} \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}-\frac {3\ 3^{2/3} \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (6 x^2+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}+\frac {3^{2/3} \left (1+i \sqrt {3}\right ) \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (6 x^2+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}+\frac {3^{2/3} \left (1-i \sqrt {3}\right ) \left (3 x^4-8\right )}{\sqrt [3]{-9 x^6+108 x^2+4 \sqrt {3} \sqrt {-27 x^8-54 c_1 x^6+207 x^4+648 c_1 x^2+32+432 c_1{}^2}+144 c_1}}\right ) \\ y(x)\to -\frac {1}{\sqrt {3}} \\ y(x)\to \frac {1}{\sqrt {3}} \\ \end{align*}