Internal problem ID [9083]
Internal file name [OUTPUT/8018_Monday_June_06_2022_01_17_34_AM_82061435/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 749.
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.
\[ \boxed {y^{\prime }-\frac {\left (-y+x \right )^{2} \left (y+x \right )^{2} x}{y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{4} x -2 x^{3} y^{2}+x^{5}-y^{\prime } y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-y^{4} x +2 x^{3} y^{2}-x^{5}}{y} \end {array} \]
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 -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -4*y(x)*x^6-(4*x^4-1)*(diff(y(x), 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) Reducible group (found another exponential solution) <- Kovacics algorithm successful <- differential order: 1; linearization to 2nd order successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 186
dsolve(diff(y(x),x) = (x-y(x))^2*(x+y(x))^2*x/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right ) \left (c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+\left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right )}}{c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}} \\ y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right ) \left (c_{1} \left (x^{2}+1\right ) {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+\left (x^{2}-1\right ) {\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}\right )}}{c_{1} {\mathrm e}^{-\frac {\left (x^{2}+1\right )^{2}}{2}}+{\mathrm e}^{-\frac {x^{2} \left (x^{2}-2\right )}{2}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 18.166 (sec). Leaf size: 102
DSolve[y'[x] == (x*(x - y[x])^2*(x + y[x])^2)/y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {x^2+\left (x^2-1\right ) e^{2 x^2+4 c_1}+1}}{\sqrt {1+e^{2 x^2+4 c_1}}} \\ y(x)\to \frac {\sqrt {x^2+\left (x^2-1\right ) e^{2 x^2+4 c_1}+1}}{\sqrt {1+e^{2 x^2+4 c_1}}} \\ \end{align*}