Internal problem ID [8318]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 738.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {2 a}{-y x^{2}+2 y^{4} a \,x^{2}-16 y^{2} a^{2} x +32 a^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 1096
dsolve(diff(y(x),x) = 2*a/(-x^2*y(x)+2*a*y(x)^4*x^2-16*a^2*x*y(x)^2+32*a^3),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}{12 x c_{1} a}+\frac {192 a^{3} c_{1}^{2}+x}{12 c_{1} a \left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}-\frac {1}{12 c_{1} a} \\ y \relax (x ) = -\frac {\left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}{24 x c_{1} a}-\frac {192 a^{3} c_{1}^{2}+x}{24 c_{1} a \left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}-\frac {1}{12 c_{1} a}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}{12 x c_{1} a}-\frac {192 a^{3} c_{1}^{2}+x}{12 c_{1} a \left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}{24 x c_{1} a}-\frac {192 a^{3} c_{1}^{2}+x}{24 c_{1} a \left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}-\frac {1}{12 c_{1} a}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}{12 x c_{1} a}-\frac {192 a^{3} c_{1}^{2}+x}{12 c_{1} a \left (\left (-216 c_{1}^{3} a^{2} x +576 a^{3} c_{1}^{2}+12 \sqrt {-\frac {3 \left (16384 a^{7} c_{1}^{4}-108 a^{2} x^{3} c_{1}^{4}+576 a^{3} x^{2} c_{1}^{3}-512 a^{4} x c_{1}^{2}-x^{3} c_{1}+4 x^{2} a \right )}{x}}\, c_{1} a -x \right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.325 (sec). Leaf size: 1161
DSolve[y'[x] == (2*a)/(32*a^3 - x^2*y[x] - 16*a^2*x*y[x]^2 + 2*a*x^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {2 \sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}{x}+\frac {32 a^2 (12 a+x)+16 a e^{c_1} x+2 e^{2 c_1} x}{\sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}-2 \left (4 a+e^{c_1}\right )}{24 a} \\ y(x)\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}{x}-\frac {2 i \left (\sqrt {3}-i\right ) \left (16 a^2 (12 a+x)+8 a e^{c_1} x+e^{2 c_1} x\right )}{\sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}-4 \left (4 a+e^{c_1}\right )}{48 a} \\ y(x)\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) \sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}{x}+\frac {2 i \left (\sqrt {3}+i\right ) \left (16 a^2 (12 a+x)+8 a e^{c_1} x+e^{2 c_1} x\right )}{\sqrt [3]{2304 a^4 x^2+64 a^3 x^2 \left (-x+9 e^{c_1}\right )-24 a^2 \left (9+2 e^{c_1}\right ) x^3+\sqrt {x^3 \left (x \left (-2304 a^4-64 a^3 \left (-x+9 e^{c_1}\right )+24 a^2 \left (9+2 e^{c_1}\right ) x+12 a e^{2 c_1} x+e^{3 c_1} x\right ){}^2-\left (192 a^3+x \left (4 a+e^{c_1}\right ){}^2\right ){}^3\right )}-12 a e^{2 c_1} x^3-e^{3 c_1} x^3}}-4 \left (4 a+e^{c_1}\right )}{48 a} \\ \end{align*}