Internal problem ID [9223]
Internal file name [OUTPUT/8159_Monday_June_06_2022_01_59_04_AM_26672392/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 890.
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.
\[ \boxed {y^{\prime }-\frac {x}{-y+1+y^{4}+2 y^{2} x^{2}+x^{4}+y^{6}+3 y^{4} x^{2}+3 y^{2} x^{4}+x^{6}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{6}+3 y^{\prime } y^{4} x^{2}+3 y^{\prime } y^{2} x^{4}+y^{\prime } x^{6}+y^{\prime } y^{4}+2 y^{\prime } y^{2} x^{2}+y^{\prime } x^{4}-y^{\prime } y+y^{\prime }-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{-y+1+y^{4}+2 y^{2} x^{2}+x^{4}+y^{6}+3 y^{4} x^{2}+3 y^{2} x^{4}+x^{6}} \end {array} \]
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`[-y/x, 1]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 504
dsolve(diff(y(x),x) = x/(-y(x)+1+y(x)^4+2*x^2*y(x)^2+x^4+y(x)^6+3*x^2*y(x)^4+3*x^4*y(x)^2+x^6),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {-6 x^{2} \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-2 \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {-6 x^{2} \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-2 \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-4}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {\left (-12 x^{2}-4\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-4 i \sqrt {3}+\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}+4}}{12 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ -y \left (x \right )+\frac {\left (\int _{}^{y \left (x \right )^{2}+x^{2}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}+1}d \textit {\_a} \right )}{2}-c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.142 (sec). Leaf size: 103
DSolve[y'[x] == x/(1 + x^4 + x^6 - y[x] + 2*x^2*y[x]^2 + 3*x^4*y[x]^2 + y[x]^4 + 3*x^2*y[x]^4 + y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [y(x)-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^3+3 \text {$\#$1}^2 y(x)^2+\text {$\#$1}^2+3 \text {$\#$1} y(x)^4+2 \text {$\#$1} y(x)^2+y(x)^6+y(x)^4+1\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2+6 \text {$\#$1} y(x)^2+2 \text {$\#$1}+3 y(x)^4+2 y(x)^2}\&\right ]=c_1,y(x)\right ] \]