Internal problem ID [9297]
Internal file name [OUTPUT/8233_Monday_June_06_2022_02_24_59_AM_69611839/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 964.
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.
\[ \boxed {y^{\prime }+\frac {8 x \left (a -1\right ) \left (a +1\right )}{8-8 y+x^{6}-2 y^{4} a^{2}-8 a^{2}+y^{6}+2 y^{4}+2 x^{4}+3 y^{2} x^{4}-y^{6} a^{2}-8 y^{2} a^{2} x^{2}+4 y^{2} x^{2}+3 y^{4} x^{2}-6 a^{2} x^{4}-2 a^{6} x^{4}+6 a^{4} x^{4}+a^{8} x^{6}-9 y^{2} a^{2} x^{4}-6 y^{4} a^{2} x^{2}-4 a^{6} x^{6}-4 a^{2} x^{6}+6 a^{4} x^{6}+4 y^{2} a^{4} x^{2}+3 a^{4} y^{4} x^{2}-3 a^{6} y^{2} x^{4}+9 y^{2} a^{4} x^{4}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 8 x -8 y^{\prime }-4 y^{\prime } y^{2} x^{2}+8 y^{\prime } y-3 y^{\prime } y^{4} x^{2}-8 a^{2} x -2 y^{\prime } x^{4}-4 x^{2} y^{2} y^{\prime } a^{4}-y^{\prime } x^{6}-2 y^{\prime } y^{4}-y^{\prime } y^{6}-3 y^{\prime } y^{2} x^{4}-6 y^{\prime } a^{4} x^{4}+6 y^{\prime } a^{2} x^{4}+2 y^{\prime } y^{4} a^{2}+2 y^{\prime } a^{6} x^{4}+8 y^{\prime } y^{2} a^{2} x^{2}+6 y^{\prime } y^{4} a^{2} x^{2}+3 y^{2} y^{\prime } a^{6} x^{4}-3 y^{4} y^{\prime } a^{4} x^{2}-9 y^{2} y^{\prime } a^{4} x^{4}+9 y^{2} y^{\prime } a^{2} x^{4}+y^{6} y^{\prime } a^{2}+4 y^{\prime } a^{6} x^{6}-y^{\prime } a^{8} x^{6}-6 y^{\prime } a^{4} x^{6}+4 y^{\prime } a^{2} x^{6}+8 y^{\prime } a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {8 a^{2} x -8 x}{-8+8 y-x^{6}+2 y^{4} a^{2}+8 a^{2}-y^{6}-2 y^{4}-2 x^{4}-3 y^{2} x^{4}+y^{6} a^{2}+8 y^{2} a^{2} x^{2}-4 y^{2} x^{2}-3 y^{4} x^{2}+6 a^{2} x^{4}+2 a^{6} x^{4}-6 a^{4} x^{4}-a^{8} x^{6}+9 y^{2} a^{2} x^{4}+6 y^{4} a^{2} x^{2}+4 a^{6} x^{6}+4 a^{2} x^{6}-6 a^{4} x^{6}-4 y^{2} a^{4} x^{2}-3 a^{4} y^{4} x^{2}+3 a^{6} y^{2} x^{4}-9 y^{2} a^{4} x^{4}} \end {array} \]
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, a^2-1]
✓ Solution by Maple
Time used: 1.469 (sec). Leaf size: 575
dsolve(diff(y(x),x) = -8*x*(a-1)*(a+1)/(8-a^2*y(x)^6+a^8*x^6-4*a^6*x^6+6*a^4*x^6-2*a^2*y(x)^4-2*a^6*x^4+6*a^4*x^4-6*a^2*x^4-4*a^2*x^6+y(x)^6+x^6-8*a^2+3*a^4*y(x)^4*x^2-3*a^6*y(x)^2*x^4+9*y(x)^2*a^4*x^4-9*y(x)^2*a^2*x^4+4*a^4*y(x)^2*x^2-6*y(x)^4*a^2*x^2-8*y(x)^2*a^2*x^2+3*x^2*y(x)^4+3*x^4*y(x)^2+4*x^2*y(x)^2+2*x^4+2*y(x)^4-8*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {\left (3 a^{2} x^{2}-3 x^{2}-2\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{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 (3 a^{2} x^{2}-3 x^{2}-2\right ) \left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}-\left (116+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{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 {\left (-4+\left (6 a^{2}-6\right ) x^{2}\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}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\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}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\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}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \sqrt {3}\, \sqrt {\left (-4+\left (6 a^{2}-6\right ) x^{2}\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}}{6 \left (3 \sqrt {3}+\sqrt {31}\right )^{\frac {1}{3}}} \\ \frac {4 \left (\munderset {\textit {\_R} &=\operatorname {RootOf}\left (\textit {\_Z}^{3}+2 \textit {\_Z}^{2}+8\right )}{\sum }\frac {\ln \left (-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}-\textit {\_R} \right )}{\textit {\_R} \left (3 \textit {\_R} +4\right )}\right )+\left (a^{2}-1\right ) y \left (x \right )-c_{1} a^{4}+2 a^{2} c_{1} -c_{1}}{a^{4}-2 a^{2}+1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.937 (sec). Leaf size: 264
DSolve[y'[x] == (-8*(-1 + a)*(1 + a)*x)/(8 - 8*a^2 + 2*x^4 - 6*a^2*x^4 + 6*a^4*x^4 - 2*a^6*x^4 + x^6 - 4*a^2*x^6 + 6*a^4*x^6 - 4*a^6*x^6 + a^8*x^6 - 8*y[x] + 4*x^2*y[x]^2 - 8*a^2*x^2*y[x]^2 + 4*a^4*x^2*y[x]^2 + 3*x^4*y[x]^2 - 9*a^2*x^4*y[x]^2 + 9*a^4*x^4*y[x]^2 - 3*a^6*x^4*y[x]^2 + 2*y[x]^4 - 2*a^2*y[x]^4 + 3*x^2*y[x]^4 - 6*a^2*x^2*y[x]^4 + 3*a^4*x^2*y[x]^4 + y[x]^6 - a^2*y[x]^6),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {y(x)}{(a-1) (a+1)}-\frac {8 \text {RootSum}\left [-\text {$\#$1}^3 a^6+3 \text {$\#$1}^3 a^4-3 \text {$\#$1}^3 a^2+\text {$\#$1}^3+3 \text {$\#$1}^2 a^4 y(x)^2+2 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2 y(x)^2-4 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2 y(x)^2+2 \text {$\#$1}^2-3 \text {$\#$1} a^2 y(x)^4-4 \text {$\#$1} a^2 y(x)^2+3 \text {$\#$1} y(x)^4+4 \text {$\#$1} y(x)^2+y(x)^6+2 y(x)^4+8\&,\frac {\log \left (x^2-\text {$\#$1}\right )}{3 \text {$\#$1}^2 a^4-6 \text {$\#$1}^2 a^2+3 \text {$\#$1}^2-6 \text {$\#$1} a^2 y(x)^2-4 \text {$\#$1} a^2+6 \text {$\#$1} y(x)^2+4 \text {$\#$1}+3 y(x)^4+4 y(x)^2}\&\right ]}{(a-1) (a+1) \left (2-2 a^2\right )}=c_1,y(x)\right ] \]