Internal problem ID [8593]
Internal file name [OUTPUT/7526_Sunday_June_05_2022_11_03_14_PM_63908968/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 257.
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, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {x \left (y x +x^{4}-1\right ) y^{\prime }-y \left (y x -x^{4}-1\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (y x +x^{4}-1\right ) y^{\prime }-y \left (y x -x^{4}-1\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \left (y x -x^{4}-1\right )}{x \left (y x +x^{4}-1\right )} \end {array} \]
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 trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 98
dsolve(x*(x*y(x)+x^4-1)*diff(y(x),x)-y(x)*(x*y(x)-x^4-1)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (-2 \textit {\_Z} \,x^{4} {\mathrm e}^{2 \textit {\_Z}}+2 x^{4} {\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} c_{1} x^{4}+{\mathrm e}^{2 \textit {\_Z}}-2 c_{1} {\mathrm e}^{\textit {\_Z}}+c_{1}^{2}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-2 \textit {\_Z} \,x^{4} {\mathrm e}^{2 \textit {\_Z}}+2 x^{4} {\mathrm e}^{2 \textit {\_Z}}-2 \,{\mathrm e}^{\textit {\_Z}} c_{1} x^{4}+{\mathrm e}^{2 \textit {\_Z}}-2 c_{1} {\mathrm e}^{\textit {\_Z}}+c_{1}^{2}\right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.297 (sec). Leaf size: 39
DSolve[x*(x*y[x]+x^4-1)*y'[x]-y[x]*(x*y[x]-x^4-1)==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [2 x^2+\frac {y(x)}{x}+\frac {x \left (-2 \log \left (\frac {1}{1-x y(x)}\right )-2+c_1\right )}{y(x)}=0,y(x)\right ] \]