Internal problem ID [9242]
Internal file name [OUTPUT/8178_Monday_June_06_2022_02_03_16_AM_1366197/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x^{3}+x^{3} y^{4}+2 y^{2} x^{2}+x +y^{6} x^{3}+3 y^{4} x^{2}+3 y^{2} x +1}{x^{5} y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{6} x^{3}+x^{3} y^{4}-y^{\prime } x^{5} y+3 y^{4} x^{2}+2 y^{2} x^{2}+3 y^{2} x +x^{3}+x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-x^{3}-x^{3} y^{4}-2 y^{2} x^{2}-x -y^{6} x^{3}-3 y^{4} x^{2}-3 y^{2} x -1}{y x^{5}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 2 `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[2*x^2, 1/y]
✓ 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]
\[ \text {Solve}\left [\frac {1}{2} \text {RootSum}\left [2 \text {$\#$1}^3+2 \text {$\#$1}^2+1\&,\frac {\log \left (\frac {x y(x)^2+1}{x}-\text {$\#$1}\right )}{3 \text {$\#$1}^2+2 \text {$\#$1}}\&\right ]+\frac {1}{x}+c_1=0,y(x)\right ] \]