Internal problem ID [7907]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 327.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (y^{4} x +2 x^{2} y^{3}+2 y+x \right ) y^{\prime }+y^{5}+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 583
dsolve((x*y(x)^4+2*x^2*y(x)^3+2*y(x)+x)*diff(y(x),x)+y(x)^5+y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+4 x^{4} c_{1}+18 x^{2} c_{1}^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}{6 c_{1} x}+\frac {-6 c_{1} x^{2}+2}{3 \left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+4 x^{4} c_{1}+18 x^{2} c_{1}^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} c_{1} x}-\frac {1}{3 c_{1} x} \\ y \relax (x ) = \frac {\left (-12 i x^{2} c_{1}-i \left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}+4 i\right ) \sqrt {3}+12 c_{1} x^{2}-\left (\left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )^{2}}{12 \left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} x c_{1}} \\ y \relax (x ) = \frac {\left (12 i x^{2} c_{1}+i \left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}-4 i\right ) \sqrt {3}+12 c_{1} x^{2}-\left (\left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )^{2}}{12 \left (108 x^{2} c_{1}^{3}+12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{4}+18 x^{2} c_{1}^{2}+\left (4 x^{4}-4\right ) c_{1}-x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} x c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 17.934 (sec). Leaf size: 658
DSolve[y[x] + y[x]^5 + (x + 2*y[x] + 2*x^2*y[x]^3 + x*y[x]^4)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {2 c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x} \\ y(x)\to \frac {-\frac {4 \sqrt [3]{-2} c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+2 c_1{}^3}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\ y(x)\to \frac {\frac {4 (-1)^{2/3} c_1 \left (3 x^2+c_1\right )}{\sqrt [3]{\frac {9}{2} \left (3+c_1{}^2\right ) x^2+\frac {3}{2} \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+c_1{}^3}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {-12 c_1{}^3 x^6+\left (81-3 c_1{}^4+54 c_1{}^2\right ) x^4+12 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\ y(x)\to 0 \\ y(x)\to -\sqrt [4]{-1} \\ y(x)\to \sqrt [4]{-1} \\ y(x)\to -(-1)^{3/4} \\ y(x)\to (-1)^{3/4} \\ y(x)\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right ) \\ y(x)\to \frac {1}{2} x \left (-1+\frac {i x^2}{\sqrt {-x^4}}\right ) \\ y(x)\to -\frac {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \\ \end{align*}