Internal problem ID [9078]
Internal file name [OUTPUT/8013_Monday_June_06_2022_01_16_50_AM_48865191/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 744.
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.
\[ \boxed {y^{\prime }-\frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{4}+2 y^{\prime } y^{2} x^{2}+y^{\prime } x^{4}-y^{\prime } y-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x}{-y+x^{4}+2 x^{2} y^{2}+y^{4}} \end {array} \]
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) = -(4*x^4-4*x)*y(x)+4*x^2*(diff(y(x), x)), y(x)` *** Sublevel 2 *** 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: 0.0 (sec). Leaf size: 511
dsolve(diff(y(x),x) = x/(-y(x)+x^4+2*x^2*y(x)^2+y(x)^4),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 c_{1} x^{2}+81}\right )^{\frac {1}{3}}}{6}+\frac {c_{1}^{2}-12 x^{2}}{6 \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {3 c_{1}^{4} x^{2}+24 c_{1}^{2} x^{4}+48 x^{6}+3 c_{1}^{3}+108 c_{1} x^{2}+81}\right )^{\frac {1}{3}}}-\frac {c_{1}}{6} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}+\frac {c_{1} \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\left (i \sqrt {3}-1\right ) \left (x^{2}-\frac {c_{1}^{2}}{12}\right )}{\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {2}{3}}}{12}-\frac {c_{1} \left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\left (1+i \sqrt {3}\right ) \left (x^{2}-\frac {c_{1}^{2}}{12}\right )}{\left (-36 c_{1} x^{2}-54-c_{1}^{3}+6 \sqrt {48 x^{6}+24 c_{1}^{2} x^{4}+\left (3 c_{1}^{4}+108 c_{1} \right ) x^{2}+3 c_{1}^{3}+81}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 16.665 (sec). Leaf size: 564
DSolve[y'[x] == x/(x^4 - y[x] + 2*x^2*y[x]^2 + y[x]^4),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{6 \sqrt [3]{2}}+\frac {2^{2/3} \left (-3 x^2+c_1{}^2\right )}{3 \sqrt [3]{36 c_1 x^2+3 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-27+4 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to \frac {\left (-1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{144 c_1 x^2+2 \sqrt {\left (12 x^2-4 c_1{}^2\right ){}^3+4 \left (36 c_1 x^2-27+4 c_1{}^3\right ){}^2}-108+16 c_1{}^3}}{12 \sqrt [3]{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (3 x^2-c_1{}^2\right )}{3 \sqrt [3]{72 c_1 x^2+6 \sqrt {3} \sqrt {16 x^6+32 c_1{}^2 x^4+8 c_1 \left (-9+2 c_1{}^3\right ) x^2+27-8 c_1{}^3}-54+8 c_1{}^3}}+\frac {c_1}{3} \\ y(x)\to -i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2} \\ \end{align*}