Internal problem ID [9166]
Internal file name [OUTPUT/8101_Monday_June_06_2022_01_45_42_AM_70264924/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 832.
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.
\[ \boxed {y^{\prime }-\frac {\left (x +y+1\right ) y}{\left (y^{4}+y^{3}+y^{2}+x \right ) \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{4} x +y^{\prime } y^{4}+y^{\prime } y^{3} x +y^{\prime } y^{3}+y^{\prime } y^{2} x +y^{\prime } y^{2}+y^{\prime } x^{2}+y^{\prime } x -y^{2}-y x -y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2}+y x +y}{y^{4} x +y^{4}+y^{3} x +y^{3}+y^{2} x +y^{2}+x^{2}+x} \end {array} \]
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`[0, (x*y^2+y^2)/(y^4+y^3+y^2+x)/(x+1)]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
dsolve(diff(y(x),x) = 1/(y(x)^4+y(x)^3+y(x)^2+x)*(x+y(x)+1)*y(x)/(x+1),y(x), singsol=all)
\[ \ln \left (x +1\right )+\frac {x}{y \left (x \right )}-\frac {y \left (x \right )^{3}}{3}-\frac {y \left (x \right )^{2}}{2}-y \left (x \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 61.363 (sec). Leaf size: 2405
DSolve[y'[x] == (y[x]*(1 + x + y[x]))/((1 + x)*(x + y[x]^2 + y[x]^3 + y[x]^4)),y[x],x,IncludeSingularSolutions -> True]
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