Internal problem ID [8322]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 744.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }-\frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 615
dsolve(diff(y(x),x) = x/(-y(x)+x^4+2*x^2*y(x)^2+y(x)^4),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 x^{2} c_{1} +81}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}^{2}-12 x^{2}}{6 \left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 x^{2} c_{1} +81}\right )^{\frac {1}{3}}}-\frac {c_{1}}{6} \\ y \left (x \right ) = -\frac {2 c_{1} \left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}+i \left (\left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}+12 x^{2}-c_{1}^{2}\right ) \sqrt {3}+\left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}-12 x^{2}+c_{1}^{2}}{12 \left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}} \\ y \left (x \right ) = \frac {-2 c_{1} \left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}+i \left (\left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}+12 x^{2}-c_{1}^{2}\right ) \sqrt {3}-\left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}+12 x^{2}-c_{1}^{2}}{12 \left (-36 x^{2} c_{1} -54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 17.009 (sec). Leaf size: 564
DSolve[y'[x] == x/(x^4 - y[x] + 2*x^2*y[x]^2 + y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{6 \sqrt [3]{2}}+\frac {2^{2/3} \left (-3 x^2+c_1{}^2\right )}{3 \sqrt [3]{36 c_1 x^2+3 \left (-9+\sqrt {81+24 \left (2 x^6+4 c_1{}^2 x^4+c_1 \left (-9+2 c_1{}^3\right ) x^2-c_1{}^3\right )}\right )+4 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{36 c_1 x^2+3 \left (-9+\sqrt {81+24 \left (2 x^6+4 c_1{}^2 x^4+c_1 \left (-9+2 c_1{}^3\right ) x^2-c_1{}^3\right )}\right )+4 c_1{}^3}}{6\ 2^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \left (-9+\sqrt {81+24 \left (2 x^6+4 c_1{}^2 x^4+c_1 \left (-9+2 c_1{}^3\right ) x^2-c_1{}^3\right )}\right )+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \left (-9+\sqrt {81+24 \left (2 x^6+4 c_1{}^2 x^4+c_1 \left (-9+2 c_1{}^3\right ) x^2-c_1{}^3\right )}\right )+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2} \\ \end{align*}