Internal problem ID [9217]
Internal file name [OUTPUT/8153_Monday_June_06_2022_01_57_53_AM_23581245/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 884.
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.
\[ \boxed {y^{\prime }+\frac {\left (-1-y^{4}+2 y^{2} x^{2}-x^{4}-y^{6}+3 y^{4} x^{2}-3 y^{2} x^{4}+x^{6}\right ) x}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{6} x +3 y^{4} x^{3}-3 y^{2} x^{5}+x^{7}-y^{4} x +2 x^{3} y^{2}-x^{5}+y^{\prime } y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{6} x -3 y^{4} x^{3}+3 y^{2} x^{5}-x^{7}+y^{4} x -2 x^{3} y^{2}+x^{5}+x}{y} \end {array} \]
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`[0, (-x^6+3*x^4*y^2-3*x^2*y^4+y^6+x^4-2*x^2*y^2+y^4)/y]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 106
dsolve(diff(y(x),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)*x/y(x),y(x), singsol=all)
\[ y \left (x \right ) = -{\mathrm e}^{\operatorname {RootOf}\left (-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 x^{3} {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{2 \textit {\_Z}} \ln \left (\frac {{\mathrm e}^{2 \textit {\_Z}}-2 x \,{\mathrm e}^{\textit {\_Z}}+1}{{\mathrm e}^{\textit {\_Z}}-2 x}\right )-2 \,{\mathrm e}^{2 \textit {\_Z}} c_{1} -\textit {\_Z} \,{\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (\frac {{\mathrm e}^{2 \textit {\_Z}}-2 x \,{\mathrm e}^{\textit {\_Z}}+1}{{\mathrm e}^{\textit {\_Z}}-2 x}\right ) x +4 c_{1} x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} x \,{\mathrm e}^{\textit {\_Z}}-1\right )}+x \]
✓ Solution by Mathematica
Time used: 0.475 (sec). Leaf size: 71
DSolve[y'[x] == (x*(1 + x^4 - x^6 - 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],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{4} \left (2 \log \left (-x^2+y(x)^2+1\right )-2 x^2-\frac {1}{y(x) (y(x)+x)}+\frac {1}{x y(x)-y(x)^2}-2 \log (x-y(x))-2 \log (y(x)+x)\right )=c_1,y(x)\right ] \]