Internal problem ID [9121]
Internal file name [OUTPUT/8056_Monday_June_06_2022_01_27_23_AM_50937649/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 787.
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, [_1st_order, `_with_symmetry_[F(x),G(x)]`], [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {x \left (-x -1+x^{2}-2 y x^{2}+2 x^{4}\right )}{\left (x^{2}-y\right ) \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{5}-y^{\prime } x^{3}-2 y x^{3}+y^{\prime } x y-x^{2} y^{\prime }+x^{3}+y y^{\prime }-x^{2}-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x^{5}+2 y x^{3}-x^{3}+x^{2}+x}{-x^{3}+y x -x^{2}+y} \end {array} \]
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: 191
dsolve(diff(y(x),x) = 1/(x^2-y(x))*x*(-x-1+x^2-2*x^2*y(x)+2*x^4)/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = \frac {4 x^{2} {\mathrm e}^{\operatorname {RootOf}\left (8 x^{3} {\mathrm e}^{\textit {\_Z}}-24 x^{2} {\mathrm e}^{\textit {\_Z}}-36 x^{3}+6 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right ) {\mathrm e}^{\textit {\_Z}}+18 c_{1} {\mathrm e}^{\textit {\_Z}}-6 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +24 x \,{\mathrm e}^{\textit {\_Z}}+108 x^{2}-27 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right )-81 c_{1} +27 \textit {\_Z} -108 x +27\right )}-18 x^{2}-9}{4 \,{\mathrm e}^{\operatorname {RootOf}\left (8 x^{3} {\mathrm e}^{\textit {\_Z}}-24 x^{2} {\mathrm e}^{\textit {\_Z}}-36 x^{3}+6 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right ) {\mathrm e}^{\textit {\_Z}}+18 c_{1} {\mathrm e}^{\textit {\_Z}}-6 \,{\mathrm e}^{\textit {\_Z}} \textit {\_Z} +24 x \,{\mathrm e}^{\textit {\_Z}}+108 x^{2}-27 \ln \left (\frac {2 \,{\mathrm e}^{\textit {\_Z}}-9}{\left (x +1\right )^{4}}\right )-81 c_{1} +27 \textit {\_Z} -108 x +27\right )}-18} \]
✓ Solution by Mathematica
Time used: 17.772 (sec). Leaf size: 488
DSolve[y'[x] == (x*(-1 - x + x^2 + 2*x^4 - 2*x^2*y[x]))/((1 + x)*(x^2 - y[x])),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (2-\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{\sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}\right ) \left (\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{\sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}+4\right ) \left (\left (1-\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{2 \sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}\right ) \log \left (\frac {2-\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{\sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}}{\sqrt [3]{2}}\right )+\left (\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{2 \sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}-1\right ) \log \left (\frac {\frac {x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{\sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}+4}{\sqrt [3]{2}}\right )-3\right )}{18 \sqrt [3]{2} \left (-\frac {\left (2 x^2-2 y(x)+3\right )^3}{8 \left (x^2-y(x)\right )^3}+\frac {3 x \left (x^2-x-1\right ) \left (2 x^2-2 y(x)+3\right )}{2 \sqrt [3]{x^3 \left (x^2-x-1\right )^3} \left (x^2-y(x)\right )}-2\right )}=\frac {4\ 2^{2/3} \left (x^3 \left (x^2-x-1\right )^3\right )^{2/3} \left (x \left (x^2-3 x+3\right )-3 \log (x+1)\right )}{27 x^2 \left (-x^2+x+1\right )^2}+c_1,y(x)\right ] \]