Internal problem ID [9250]
Internal file name [OUTPUT/8186_Monday_June_06_2022_02_08_02_AM_45314092/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 917.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {y \left (x \ln \left (x \right )+\ln \left (x \right )+\ln \left (y\right ) x +\ln \left (y\right )-x -1+\ln \left (x \right )^{2} x +2 x \ln \left (y\right ) \ln \left (x \right )+x \ln \left (y\right )^{2}\right )}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \ln \left (x \right )^{2} y x +2 x \ln \left (x \right ) y \ln \left (y\right )+\ln \left (y\right )^{2} y x +y \ln \left (x \right ) x -x^{2} y^{\prime }+\ln \left (y\right ) y x +y \ln \left (x \right )-y^{\prime } x +y \ln \left (y\right )-x y-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\ln \left (x \right )^{2} y x -2 x \ln \left (x \right ) y \ln \left (y\right )-\ln \left (y\right )^{2} y x -y \ln \left (x \right ) x -\ln \left (y\right ) y x -y \ln \left (x \right )-y \ln \left (y\right )+x y+y}{-x^{2}-x} \end {array} \]
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, (ln(x)^2+2*ln(x)*ln(y)+ln(y)^2)*y/x]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve(diff(y(x),x) = y(x)*(x*ln(x)+ln(x)+ln(y(x))*x+ln(y(x))-x-1+x*ln(x)^2+2*x*ln(y(x))*ln(x)+x*ln(y(x))^2)/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {\ln \left (x \right ) \ln \left (x +1\right )+\ln \left (x \right ) \left (-x +c_{1} \right )-x}{-\ln \left (x +1\right )-c_{1} +x}} \]
✓ Solution by Mathematica
Time used: 0.437 (sec). Leaf size: 35
DSolve[y'[x] == ((-1 - x + Log[x] + x*Log[x] + x*Log[x]^2 + Log[y[x]] + x*Log[y[x]] + 2*x*Log[x]*Log[y[x]] + x*Log[y[x]]^2)*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {x}{x-\log (x+1)-c_1}}}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}