Internal problem ID [9073]
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')`]
\[ \boxed {y^{\prime }-\frac {2 a}{-y x^{2}+2 a y^{4} x^{2}-16 a^{2} x y^{2}+32 a^{3}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 968
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 \left (x \right ) &= \frac {192 c_{1}^{2} a^{3} x +x^{2}-x \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}+\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {2}{3}}}{12 c_{1} x a \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {2}{3}}}{24}+8 x \left (-\frac {\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}}{96}+\left (i \sqrt {3}-1\right ) \left (a^{3} c_{1}^{2}+\frac {x}{192}\right )\right )}{c_{1} x a \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {2}{3}}}{24}+8 \left (-\frac {\left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}}{96}+\left (-i \sqrt {3}-1\right ) \left (a^{3} c_{1}^{2}+\frac {x}{192}\right )\right ) x}{c_{1} x a \left (-216 c_{1}^{3} a^{2} x^{3}+576 c_{1}^{2} a^{3} x^{2}+12 a c_{1} x^{2} \sqrt {\frac {\left (324 a^{2} c_{1}^{4}+3 c_{1} \right ) x^{3}+\left (-1728 a^{3} c_{1}^{3}-12 a \right ) x^{2}+1536 c_{1}^{2} a^{4} x -49152 c_{1}^{4} a^{7}}{x}}-x^{3}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.321 (sec). Leaf size: 1200
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^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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 \left (192 a^3+16 a^2 x+8 a e^{c_1} x+e^{2 c_1} x\right )}{\sqrt [3]{2304 a^4 x^2-64 a^3 x^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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 (192 a^3+16 a^2 x+8 a e^{c_1} x+e^{2 c_1} x\right )}{\sqrt [3]{2304 a^4 x^2-64 a^3 x^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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 (192 a^3+16 a^2 x+8 a e^{c_1} x+e^{2 c_1} x\right )}{\sqrt [3]{2304 a^4 x^2-64 a^3 x^3+576 a^3 e^{c_1} x^2-216 a^2 x^3-48 a^2 e^{c_1} 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*}