Internal problem ID [9105]
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]
\[ \boxed {y^{\prime }-\frac {2 y^{6}}{y^{3}+2+16 y^{2} x +32 y^{4} x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.016 (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 \left (x \right ) = \frac {\left (4096 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \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 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \right )^{\frac {1}{3}}}+\frac {16 x c_{1}}{3 \left (c_{1} +16 x \right )} y \left (x \right ) = -\frac {\left (4096 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \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 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \right )^{\frac {1}{3}}}+\frac {16 x c_{1}}{3 \left (c_{1} +16 x \right )}-\frac {i \sqrt {3}\, \left (\frac {\left (4096 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \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 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \right )^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {\left (4096 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \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 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \right )^{\frac {1}{3}}}+\frac {16 x c_{1}}{3 \left (c_{1} +16 x \right )}+\frac {i \sqrt {3}\, \left (\frac {\left (4096 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \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 x^{3} c_{1}^{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 x \,c_{1}^{2}+9216 x^{2} c_{1} \right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 27.592 (sec). Leaf size: 952
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]{2} \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}+\frac {4\ 2^{2/3} \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}}+32 x}{6 (1-16 c_1 x)} y(x)\to \frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}-\frac {4 i 2^{2/3} \left (\sqrt {3}-i\right ) \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}}+64 x}{12 (1-16 c_1 x)} y(x)\to \frac {-2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}+\frac {4 i 2^{2/3} \left (\sqrt {3}+i\right ) \left (64 x^2-48 c_1{}^2 x+3 c_1\right )}{\sqrt [3]{2048 x^3+4608 c_1{}^2 x^2+3 \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 )}-720 c_1 x+27}}+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*}