Internal problem ID [9229]
Internal file name [OUTPUT/8165_Monday_June_06_2022_02_00_16_AM_667058/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 896.
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.
\[ \boxed {y^{\prime }-\frac {x +1+y^{4}-2 y^{2} x^{2}+x^{4}+y^{6}-3 y^{4} x^{2}+3 y^{2} x^{4}-x^{6}}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y^{6}+3 y^{4} x^{2}-3 y^{2} x^{4}+x^{6}-y^{4}+2 y^{2} x^{2}-x^{4}+y^{\prime } y-x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x +1+y^{4}-2 y^{2} x^{2}+x^{4}+y^{6}-3 y^{4} x^{2}+3 y^{2} x^{4}-x^{6}}{y} \end {array} \]
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+1)/y]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 61
dsolve(diff(y(x),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)/y(x),y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}}{\textit {\_a}^{6}-3 \textit {\_a}^{4} x^{2}+3 x^{4} \textit {\_a}^{2}-x^{6}+\textit {\_a}^{4}-2 \textit {\_a}^{2} x^{2}+x^{4}+1}d \textit {\_a} \right )+x -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.205 (sec). Leaf size: 106
DSolve[y'[x] == (1 + x + 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]
\[ \text {Solve}\left [\frac {1}{2} \text {RootSum}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2 y(x)^2+\text {$\#$1}^2-3 \text {$\#$1} y(x)^4-2 \text {$\#$1} y(x)^2+y(x)^6+y(x)^4+1\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2-6 \text {$\#$1} y(x)^2-2 \text {$\#$1}+3 y(x)^4+2 y(x)^2}\&\right ]-x=c_1,y(x)\right ] \]