Internal problem ID [8350]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 770.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 y^{6}}{y^{3}+2+16 x y^{2}+32 x^{2} y^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 1345
dsolve(diff(y(x),x) = 2*y(x)^6/(y(x)^3+2+16*x*y(x)^2+32*x^2*y(x)^4),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}{3 c_{1}+48 x}+\frac {\frac {256}{3} c_{1}^{2} x^{2}-4 c_{1}-64 x}{\left (c_{1}+16 x \right ) \left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}+\frac {16 c_{1} x}{3 \left (c_{1}+16 x \right )} \\ y \relax (x ) = -\frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}{6 \left (c_{1}+16 x \right )}-\frac {2 \left (64 c_{1}^{2} x^{2}-3 c_{1}-48 x \right )}{3 \left (c_{1}+16 x \right ) \left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}+\frac {16 c_{1} x}{3 \left (c_{1}+16 x \right )}-\frac {i \sqrt {3}\, \left (\frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}{3 c_{1}+48 x}-\frac {4 \left (64 c_{1}^{2} x^{2}-3 c_{1}-48 x \right )}{3 \left (c_{1}+16 x \right ) \left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}{6 \left (c_{1}+16 x \right )}-\frac {2 \left (64 c_{1}^{2} x^{2}-3 c_{1}-48 x \right )}{3 \left (c_{1}+16 x \right ) \left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}+\frac {16 c_{1} x}{3 \left (c_{1}+16 x \right )}+\frac {i \sqrt {3}\, \left (\frac {\left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}{3 c_{1}+48 x}-\frac {4 \left (64 c_{1}^{2} x^{2}-3 c_{1}-48 x \right )}{3 \left (c_{1}+16 x \right ) \left (4096 c_{1}^{3} x^{3}+6 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, c_{1}+96 \sqrt {3}\, \sqrt {4096 c_{1}^{4} x^{3}+27 c_{1}^{4}+576 c_{1}^{3} x +2048 c_{1}^{2} x^{2}+16 c_{1}+256 x}\, x +54 c_{1}^{3}+1440 c_{1}^{2} x +9216 c_{1} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.875 (sec). Leaf size: 938
DSolve[y'[x] == (2*y[x]^6)/(2 + 16*x*y[x]^2 + y[x]^3 + 32*x^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [3]{6 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+32 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}+\frac {4\ 2^{2/3} \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{3 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+16 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}}+32 x}{6 (1-16 c_1 x)} \\ y(x)\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{6 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+32 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}-\frac {8 \sqrt [3]{-1} 2^{2/3} \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{3 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+16 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}}+64 x}{12 (1-16 c_1 x)} \\ y(x)\to \frac {-4 \sqrt [3]{-2} \sqrt [3]{3 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+16 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}+\frac {8 (-2)^{2/3} \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{3 \left (9+\sqrt {3} \sqrt {(1-16 c_1 x){}^2 \left (4096 x^3+2048 c_1{}^2 x^2+64 c_1 \left (-9+4 c_1{}^3\right ) x+27-16 c_1{}^3\right )}\right )+16 x \left (32 x \left (4 x+9 c_1{}^2\right )-45 c_1\right )}}+64 x}{12 (1-16 c_1 x)} \\ y(x)\to 0 \\ y(x)\to \frac {x-\sqrt [3]{x^3}}{2 \sqrt {3} x \sqrt [6]{x^3}} \\ y(x)\to \frac {\sqrt [3]{x^3}-x}{2 \sqrt {3} x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}-3 i\right ) x-\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3}}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}+3 i\right ) x-\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^3}}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}-3 i\right ) \sqrt [3]{x^3}-\left (\sqrt {3}+3 i\right ) x}{12 x \sqrt [6]{x^3}} \\ y(x)\to \frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{x^3}-\left (\sqrt {3}-3 i\right ) x}{12 x \sqrt [6]{x^3}} \\ \end{align*}