Internal problem ID [9243]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 909.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }-\frac {x^{3}+y^{4} x^{3}+2 y^{2} x^{2}+x +x^{3} y^{6}+3 y^{4} x^{2}+3 y^{2} x +1}{x^{5} y}=0} \]
✓ Solution by Maple
Time used: 0.906 (sec). Leaf size: 728
dsolve(diff(y(x),x) = (x^3+y(x)^4*x^3+2*x^2*y(x)^2+x+x^3*y(x)^6+3*x^2*y(x)^4+3*x*y(x)^2+1)/x^5/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {x \left (\frac {2^{\frac {2}{3}} \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {2}{3}}}{4}-\frac {2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}}{2}+x^{2}\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}}}{3 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {x \left (\frac {2^{\frac {2}{3}} \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {2}{3}}}{4}-\frac {2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}}{2}+x^{2}\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}}}{3 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {4}\, \sqrt {\left (-\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {2}{3}}}{4}-2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}}+x^{2} \left (i \sqrt {3}-1\right )\right ) x \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}}}}{12 \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {4}\, \sqrt {\left (-\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {2}{3}}}{4}-2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}}+x^{2} \left (i \sqrt {3}-1\right )\right ) x \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}}}}{12 \left (x^{3} \left (-31+3 \sqrt {105}\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {-4 x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} \left (-\frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {2}{3}}}{4}+2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}+x^{2} \left (1+i \sqrt {3}\right )\right )}}{12 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {-4 x \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} \left (-\frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {2}{3}}}{4}+2^{\frac {1}{3}} \left (x +3\right ) \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}}+x^{2} \left (1+i \sqrt {3}\right )\right )}}{12 \left (x^{3} \left (3 \sqrt {3}\, \sqrt {35}-31\right )\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {x \left (\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right ) x +c_{1} x +1\right ) x -1\right )}}{x} \\ y \left (x \right ) &= -\frac {\sqrt {x \left (\operatorname {RootOf}\left (\left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right ) x +c_{1} x +1\right ) x -1\right )}}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.113 (sec). Leaf size: 64
DSolve[y'[x] == (1 + x + x^3 + 3*x*y[x]^2 + 2*x^2*y[x]^2 + 3*x^2*y[x]^4 + x^3*y[x]^4 + x^3*y[x]^6)/(x^5*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^3+2 \text {$\#$1}^2+1\&,\frac {\log \left (\frac {x y(x)^2+1}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+2 \text {$\#$1}}\&\right ]+\frac {1}{x}+c_1=0,y(x)\right ] \]