Internal problem ID [9241]
Internal file name [OUTPUT/8177_Monday_June_06_2022_02_03_06_AM_21515574/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 908.
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.
\[ \boxed {y^{\prime }-\frac {4 x \left (a -1\right ) \left (a +1\right )}{4 y+a^{2} y^{4}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 x^{2} y^{2}-x^{4}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } a^{6} x^{4}-2 y^{\prime } y^{2} a^{4} x^{2}-3 y^{\prime } a^{4} x^{4}+y^{\prime } y^{4} a^{2}+4 y^{\prime } y^{2} a^{2} x^{2}+3 y^{\prime } a^{2} x^{4}-y^{\prime } y^{4}-2 y^{\prime } y^{2} x^{2}-y^{\prime } x^{4}-4 a^{2} x +4 y^{\prime } y+4 x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {4 a^{2} x -4 x}{4 y+a^{2} y^{4}-2 a^{4} y^{2} x^{2}+4 y^{2} a^{2} x^{2}+a^{6} x^{4}-3 a^{4} x^{4}+3 a^{2} x^{4}-y^{4}-2 x^{2} y^{2}-x^{4}} \end {array} \]
Maple trace Kovacic algorithm successful
`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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(1/4)*(a^2*x^3-x^3+4)*x*(a^2-1)*y(x)-x^2*(a^2-1)*(diff(y(x), x)), y(x)` Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful <- change of variables {x -> y(x), y(x) -> x} succesful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 1349
dsolve(diff(y(x),x) = 4*x*(a-1)*(a+1)/(4*y(x)+a^2*y(x)^4-2*a^4*y(x)^2*x^2+4*y(x)^2*a^2*x^2+a^6*x^4-3*a^4*x^4+3*a^2*x^4-y(x)^4-2*x^2*y(x)^2-x^4),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (9^{\frac {2}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {2}{3}}+\left (-a^{2} c_{1} +c_{1} \right ) 9^{\frac {1}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}}+3 a^{6} x^{2}+\left (c_{1}^{2}-9 x^{2}\right ) a^{4}+\left (-2 c_{1}^{2}+9 x^{2}\right ) a^{2}-3 x^{2}+c_{1}^{2}\right ) 9^{\frac {2}{3}}}{9 {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}} \left (3 a^{2}-3\right )} \\ y \left (x \right ) &= \frac {\left (\left (-\frac {i \sqrt {3}}{3}-\frac {1}{3}\right ) 9^{\frac {2}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {2}{3}}+\left (-\frac {2 \,9^{\frac {1}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}} c_{1}}{3}+\left (i \sqrt {3}-1\right ) \left (a +1\right ) \left (a -1\right ) \left (a^{2} x^{2}-x^{2}+\frac {1}{3} c_{1}^{2}\right )\right ) \left (a +1\right ) \left (a -1\right )\right ) 9^{\frac {2}{3}}}{3 {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}} \left (6 a^{2}-6\right )} \\ y \left (x \right ) &= -\frac {9^{\frac {2}{3}} \left (\left (-\frac {i \sqrt {3}}{3}+\frac {1}{3}\right ) 9^{\frac {2}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {2}{3}}+\left (\frac {2 \,9^{\frac {1}{3}} {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}} c_{1}}{3}+\left (1+i \sqrt {3}\right ) \left (a +1\right ) \left (a -1\right ) \left (a^{2} x^{2}-x^{2}+\frac {1}{3} c_{1}^{2}\right )\right ) \left (a +1\right ) \left (a -1\right )\right )}{3 {\left (\left (3+\frac {\sqrt {-3 \left (a -1\right )^{5} \left (a +1\right )^{5} x^{6}+6 c_{1}^{2} \left (a -1\right )^{4} \left (a +1\right )^{4} x^{4}-3 c_{1} \left (a -1\right )^{2} \left (a +1\right )^{2} \left (c_{1}^{3} a^{2}-c_{1}^{3}-18\right ) x^{2}-6 c_{1}^{3} a^{2}+6 c_{1}^{3}+81}}{3}+\frac {\left (-a^{2}+1\right ) c_{1}^{3}}{9}+x^{2} \left (a -1\right )^{2} \left (a +1\right )^{2} c_{1} \right ) \left (a +1\right )^{2} \left (a -1\right )^{2}\right )}^{\frac {1}{3}} \left (6 a^{2}-6\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 9.455 (sec). Leaf size: 1065
DSolve[y'[x] == (4*(-1 + a)*(1 + a)*x)/(-x^4 + 3*a^2*x^4 - 3*a^4*x^4 + a^6*x^4 + 4*y[x] - 2*x^2*y[x]^2 + 4*a^2*x^2*y[x]^2 - 2*a^4*x^2*y[x]^2 - y[x]^4 + a^2*y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}+\frac {3 \left (a^2-1\right )^3 x^2+c_1{}^2}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+c_1}{3 \left (a^2-1\right )} \\ y(x)\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}-\frac {2 i \left (\sqrt {3}-i\right ) \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right )}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+4 c_1}{12 \left (a^2-1\right )} \\ y(x)\to \frac {-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}+\frac {2 i \left (\sqrt {3}+i\right ) \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right )}{\sqrt [3]{-9 a^6 c_1 x^2+27 a^4 c_1 x^2+27 a^4-27 a^2 c_1 x^2-54 a^2+\frac {1}{2} \sqrt {4 \left (-9 a^6 c_1 x^2+27 a^4 \left (1+c_1 x^2\right )-27 a^2 \left (2+c_1 x^2\right )+9 c_1 x^2+27+c_1{}^3\right ){}^2-4 \left (3 \left (a^2-1\right )^3 x^2+c_1{}^2\right ){}^3}+9 c_1 x^2+27+c_1{}^3}}+4 c_1}{12 \left (a^2-1\right )} \\ y(x)\to -\frac {i \sqrt {-\left (a^2-1\right )^3 x^2}}{a^2-1} \\ y(x)\to \frac {i \sqrt {-\left (a^2-1\right )^3 x^2}}{a^2-1} \\ \end{align*}