Internal problem ID [9058]
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]]
\[ \boxed {y^{\prime }-\frac {2 a}{y+2 y^{4} a -16 a^{2} x y^{2}+32 a^{3} x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.031 (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 \left (x \right ) = \frac {{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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} a x -\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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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 c_{1} a}{3} y \left (x \right ) = -\frac {{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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} a x -\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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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 c_{1} a}{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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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} a x -\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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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 \left (x \right ) = -\frac {{\left (\left (64 c_{1}^{3} a^{4}-576 c_{1} a^{3} x +3 \sqrt {-12288 a^{7} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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} a x -\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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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 c_{1} a}{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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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} a x -\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} c_{1}^{4} x +24576 a^{6} c_{1}^{2} x^{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: 19.546 (sec). Leaf size: 672
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 {\sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{12 \sqrt [3]{2} a}-\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 {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\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 (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1+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 {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\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 (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1-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 {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3} \end{align*}