Internal problem ID [7898]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 318.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]
Solve \begin {gather*} \boxed {\left (3 x y^{3}-4 y x +y\right ) y^{\prime }+y^{2} \left (y^{2}-2\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 28
dsolve((3*x*y(x)^3-4*x*y(x)+y(x))*diff(y(x),x)+y(x)^2*(y(x)^2-2) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ x +\frac {1}{y \relax (x )^{2}}-\frac {c_{1}}{\sqrt {y \relax (x )^{2}-2}\, y \relax (x )^{2}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 58.222 (sec). Leaf size: 1942
DSolve[y[x]^2*(-2 + y[x]^2) + (y[x] - 4*x*y[x] + 3*x*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 0 \\ y(x)\to -\frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3-4 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}+2 x^2 \left (\sqrt [3]{2}+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{\sqrt {6}} \\ y(x)\to \frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3-4 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}+2 x^2 \left (\sqrt [3]{2}+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{\sqrt {6}} \\ y(x)\to -\frac {\sqrt {\frac {4 x^2 \left (\text {Root}\left [\text {$\#$1}^3-2\&,3\right ]+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )+16 (-1)^{2/3} \sqrt [3]{2} x^4+16 (-1)^{2/3} \sqrt [3]{2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (-1-i \sqrt {3}\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\frac {4 x^2 \left (\text {Root}\left [\text {$\#$1}^3-2\&,3\right ]+2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )+16 (-1)^{2/3} \sqrt [3]{2} x^4+16 (-1)^{2/3} \sqrt [3]{2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+\left (-1-i \sqrt {3}\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to -\frac {\sqrt {\frac {-16 \sqrt [3]{-2} x^4-16 \sqrt [3]{-2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+i \left (\sqrt {3}+i\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}-4 x^2 \left (\sqrt [3]{-2}-2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to \frac {\sqrt {\frac {-16 \sqrt [3]{-2} x^4-16 \sqrt [3]{-2} x^3-8 x \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}+i \left (\sqrt {3}+i\right ) \left (6 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}-54 c_1{}^2 x^4+4 (2 x+1)^3 x^3\right ){}^{2/3}-4 x^2 \left (\sqrt [3]{-2}-2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}\right )}{x^2 \sqrt [3]{3 \sqrt {3} \sqrt {c_1{}^2 x^7 \left (-4 (2 x+1)^3+27 c_1{}^2 x\right )}+x^3 \left (2 (2 x+1)^3-27 c_1{}^2 x\right )}}}}{2 \sqrt {3}} \\ y(x)\to 0 \\ y(x)\to -\sqrt {2} \\ y(x)\to \sqrt {2} \\ \end{align*}