Internal problem ID [8303]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 723.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 a}{y+2 a y^{4}-16 y^{2} a^{2} x +32 a^{3} x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 864
dsolve(diff(y(x),x) = 2*a/(y(x)+2*a*y(x)^4-16*a^2*x*y(x)^2+32*a^3*x^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}-\frac {6 \left (-\frac {4}{3} x a -\frac {4}{9} c_{1}^{2} a^{2}\right ) a}{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}+\frac {2 a c_{1}}{3} \\ y \relax (x ) = -\frac {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}{12 a}+\frac {3 \left (-\frac {4}{3} x a -\frac {4}{9} c_{1}^{2} a^{2}\right ) a}{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}+\frac {2 a c_{1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}+\frac {6 \left (-\frac {4}{3} x a -\frac {4}{9} c_{1}^{2} a^{2}\right ) a}{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}{12 a}+\frac {3 \left (-\frac {4}{3} x a -\frac {4}{9} c_{1}^{2} a^{2}\right ) a}{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}+\frac {2 a c_{1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}+\frac {6 \left (-\frac {4}{3} x a -\frac {4}{9} c_{1}^{2} a^{2}\right ) a}{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} x c_{1}^{4}+24576 a^{6} x^{2} c_{1}^{2}-12288 a^{5} x^{3}+384 c_{1}^{3} a^{4}-3456 c_{1} a^{3} x +81}+27\right ) a^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 21.011 (sec). Leaf size: 663
DSolve[y'[x] == (2*a)/(32*a^3*x^2 + y[x] - 16*a^2*x*y[x]^2 + 2*a*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {8 a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {81 a^4-384 a^7 \left (32 a^4 c_1{}^4 x-64 a^3 c_1{}^2 x^2+32 a^2 x^3-a c_1{}^3+9 c_1 x\right )}}}-\frac {\sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {81 a^4-384 a^7 \left (32 a^4 c_1{}^4 x-64 a^3 c_1{}^2 x^2+32 a^2 x^3-a c_1{}^3+9 c_1 x\right )}}}{6 a}+\frac {2 a c_1}{3} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (27+64 a^3 c_1 \left (-9 x+a c_1{}^2\right )\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {\left (4+4 i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {81 a^4-384 a^7 \left (32 a^4 c_1{}^4 x-64 a^3 c_1{}^2 x^2+32 a^2 x^3-a c_1{}^3+9 c_1 x\right )}}}+\frac {2 a c_1}{3} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (27+64 a^3 c_1 \left (-9 x+a c_1{}^2\right )\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {\left (4-4 i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {81 a^4-384 a^7 \left (32 a^4 c_1{}^4 x-64 a^3 c_1{}^2 x^2+32 a^2 x^3-a c_1{}^3+9 c_1 x\right )}}}+\frac {2 a c_1}{3} \\ \end{align*}