Internal problem ID [8663]
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]
\[ \boxed {\left (y^{4} x +2 y^{3} x^{2}+2 y+x \right ) y^{\prime }+y^{5}+y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 633
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 \left (x \right ) &= \frac {-1+\frac {\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}{2}-\frac {2 \left (3 c_{1} x^{2}-1\right )}{\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}}{3 c_{1} x} \\ y \left (x \right ) &= \frac {i \left (4-12 c_{1} x^{2}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}\right ) \sqrt {3}+12 c_{1} x^{2}-{\left (\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{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 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{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 \left (x \right ) &= \frac {12 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}+12 c_{1} x^{2}-4 i \sqrt {3}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{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 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4}{12 x c_{1} \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.131 (sec). Leaf size: 675
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 {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+c_1{}^3}}+2^{2/3} \sqrt [3]{9 \left (3+c_1{}^2\right ) x^2+3 \sqrt {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+2 c_1}{6 x} \\ y(x)\to \frac {-\frac {2 i \left (\sqrt {3}-i\right ) 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 {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^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 {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 c_1{}^3 x^2}+2 c_1{}^3}+4 c_1}{12 x} \\ y(x)\to \frac {\frac {2 i \left (\sqrt {3}+i\right ) 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 {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 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 {3} \sqrt {-4 c_1{}^3 x^6+\left (27-c_1{}^4+18 c_1{}^2\right ) x^4+4 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 {x}{2}+\frac {i \sqrt {-x^4}}{2 x} \\ \end{align*}