Internal problem ID [8489]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x^{3}+y^{4} x^{3}+2 x^{2} y^{2}+x +x^{3} y^{6}+3 x^{2} y^{4}+3 x y^{2}+1}{x^{5} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 3.053 (sec). Leaf size: 844
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 \relax (x ) = -\frac {\sqrt {6}\, \sqrt {x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-2 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {6}\, \sqrt {x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-2 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}-6 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {-3 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}+4 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}+12 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {-3 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}+4 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}+4 x^{2}+12 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\sqrt {3}\, \sqrt {x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}-\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}-4 x^{2}-12 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {3}\, \sqrt {x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}-\left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {2}{3}}-4 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}-4 x^{2}-12 \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}\right )}}{6 x \left (-62 x^{3}+6 \sqrt {105}\, x^{3}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {\sqrt {x \left (\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 \relax (x ) = -\frac {\sqrt {x \left (\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: 0.112 (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]
\[ \text {Solve}\left [\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^3+2 \text {$\#$1}^2+1\&,\frac {\log \left (\frac {x y(x)^2+1}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+2 \text {$\#$1}}\&\right ]+\frac {1}{x}+c_1=0,y(x)\right ] \]