Internal problem ID [9150]
Internal file name [OUTPUT/8085_Monday_June_06_2022_01_42_32_AM_3545567/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 816.
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.
\[ \boxed {y^{\prime }-\frac {\left (-y+x \right )^{3} \left (y+x \right )^{3} x}{\left (-y^{2}+x^{2}-1\right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{6} x -3 y^{4} x^{3}+3 y^{2} x^{5}-x^{7}-y^{\prime } y^{3}+y^{\prime } y x^{2}-y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-y^{6} x +3 y^{4} x^{3}-3 y^{2} x^{5}+x^{7}}{-y^{3}+x^{2} y-y} \end {array} \]
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^2-y^2-1)/(x^2-y^2-1)/y]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 226
dsolve(diff(y(x),x) = (x-y(x))^3*(x+y(x))^3*x/(-y(x)^2+x^2-1)/y(x),y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{x}\frac {\left (-\textit {\_a} +y \left (x \right )\right )^{3} \left (\textit {\_a} +y \left (x \right )\right )^{3} \textit {\_a}}{-\textit {\_a}^{6}+3 \textit {\_a}^{4} y \left (x \right )^{2}-3 y \left (x \right )^{4} \textit {\_a}^{2}+y \left (x \right )^{6}+\textit {\_a}^{2}-y \left (x \right )^{2}-1}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {2 \left (\left (-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\textit {\_a} -\textit {\_f} \right )^{2} \left (\textit {\_a} +\textit {\_f} \right )^{2} \textit {\_a} \left (2 \textit {\_a}^{2}-2 \textit {\_f}^{2}-3\right )}{\left (\textit {\_a}^{6}-3 \textit {\_a}^{4} \textit {\_f}^{2}+\left (3 \textit {\_f}^{4}-1\right ) \textit {\_a}^{2}-\textit {\_f}^{6}+\textit {\_f}^{2}+1\right )^{2}}d \textit {\_a} \right )+\frac {x^{2}}{2}-\frac {\textit {\_f}^{2}}{2}-\frac {1}{2}\right ) \textit {\_f}}{-\textit {\_f}^{6}+3 \textit {\_f}^{4} x^{2}+\left (-3 x^{4}+1\right ) \textit {\_f}^{2}+x^{6}-x^{2}+1}d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.193 (sec). Leaf size: 74
DSolve[y'[x] == (x*(x - y[x])^3*(x + y[x])^3)/(y[x]*(-1 + x^2 - y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}+1\&,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+x^2-y(x)^2\right )-\log \left (-\text {$\#$1}+x^2-y(x)^2\right )}{3 \text {$\#$1}^2-1}\&\right ]+x^2\right )=c_1,y(x)\right ] \]