Internal problem ID [9106]
Internal file name [OUTPUT/8041_Monday_June_06_2022_01_21_47_AM_95613725/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 772.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`x=_G(y,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (x +1+\ln \left (y\right ) x \right ) \ln \left (y\right ) y}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\ln \left (y\right )^{2} y x +x^{2} y^{\prime }-\ln \left (y\right ) y x +y^{\prime } x -y \ln \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\ln \left (y\right )^{2} y x +\ln \left (y\right ) y x +y \ln \left (y\right )}{x^{2}+x} \end {array} \]
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 = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5`[0, y*ln(y)^2/x]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 18
dsolve(diff(y(x),x) = (x+1+ln(y(x))*x)*ln(y(x))*y(x)/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{\ln \left (x +1\right )+c_{1} -x}} \]
✓ Solution by Mathematica
Time used: 0.393 (sec). Leaf size: 26
DSolve[y'[x] == (Log[y[x]]*(1 + x + x*Log[y[x]])*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {x}{-x+\log (x+1)+c_1}} \\ y(x)\to 1 \\ \end{align*}