Internal problem ID [9202]
Internal file name [OUTPUT/8138_Monday_June_06_2022_01_53_28_AM_59902064/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 869.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {-x +1-2 y+3 x^{2}-2 y x^{2}+2 x^{4}+x^{3}-2 y x^{3}+2 x^{5}}{x^{2}-y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 x^{5}-2 y x^{3}+2 x^{4}-y^{\prime } x^{2}-2 y x^{2}+x^{3}+y^{\prime } y+3 x^{2}-2 y-x +1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-2 x^{5}+2 y x^{3}-2 x^{4}+2 y x^{2}-x^{3}-3 x^{2}+2 y+x -1}{-x^{2}+y} \end {array} \]
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: 34
dsolve(diff(y(x),x) = 1/(x^2-y(x))*(-x+1-2*y(x)+3*x^2-2*x^2*y(x)+2*x^4+x^3-2*x^3*y(x)+2*x^5),y(x), singsol=all)
\[ y \left (x \right ) = x^{2}+\frac {\operatorname {LambertW}\left (-2 c_{1} {\mathrm e}^{x^{4}+\frac {4}{3} x^{3}-2 x^{2}+4 x -1}\right )}{2}+\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 3.596 (sec). Leaf size: 53
DSolve[y'[x] == (1 - x + 3*x^2 + x^3 + 2*x^4 + 2*x^5 - 2*y[x] - 2*x^2*y[x] - 2*x^3*y[x])/(x^2 - y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2+\frac {1}{2} \left (1+W\left (-e^{x^4+\frac {4 x^3}{3}-2 x^2+4 x-1+c_1}\right )\right ) \\ y(x)\to x^2+\frac {1}{2} \\ \end{align*}