Internal problem ID [9094]
Internal file name [OUTPUT/8029_Monday_June_06_2022_01_19_54_AM_9616547/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 760.
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.
\[ \boxed {y^{\prime }-\frac {\left (y^{2} x +1\right )^{3}}{x^{4} \left (y^{2} x +1+x \right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{3} x^{5}-y^{6} x^{3}+y^{\prime } y x^{5}+y^{\prime } x^{4} y-3 y^{4} x^{2}-3 y^{2} x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{6} x^{3}+3 y^{4} x^{2}+3 y^{2} x +1}{y^{3} x^{5}+x^{5} y+y x^{4}} \end {array} \]
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`[2*x^2, 1/y]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 182
dsolve(diff(y(x),x) = (x*y(x)^2+1)^3/x^4/(x*y(x)^2+1+x)/y(x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-\left (2+\left (1+i\right ) x \right ) x}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-\left (2+\left (1+i\right ) x \right ) x}}{2 x} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {x \left (-2+\left (-1+i\right ) x \right )}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {x \left (-2+\left (-1+i\right ) x \right )}}{2 x} \\ -\frac {\ln \left (2 y \left (x \right )^{4} x^{2}+\left (2 x^{2}+4 x \right ) y \left (x \right )^{2}+x^{2}+2 x +2\right )}{10}+\frac {\arctan \left (2 y \left (x \right )^{4} x +\left (2 x +2\right ) y \left (x \right )^{2}+x +1\right )}{10}+\frac {\ln \left (x y \left (x \right )^{2}-x +1\right )}{5}+\frac {1}{2 x}-\frac {\arctan \left (2 y \left (x \right )^{2}+1\right )}{10}+c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.518 (sec). Leaf size: 112
DSolve[y'[x] == (1 + x*y[x]^2)^3/(x^4*y[x]*(1 + x + x*y[x]^2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [2 \left (-\frac {1}{10} \arctan \left (2 x y(x)^4+2 x y(x)^2+2 y(x)^2+x+1\right )+\frac {1}{10} \log \left (2 x^2 y(x)^4+2 x^2 y(x)^2+x^2+4 x y(x)^2+2 x+2\right )-\frac {1}{5} \log \left (x y(x)^2-x+1\right )-\frac {1}{2 x}\right )+\frac {1}{5} \arctan \left (2 y(x)^2+1\right )=c_1,y(x)\right ] \]