Internal problem ID [9188]
Internal file name [OUTPUT/8123_Monday_June_06_2022_01_50_06_AM_43708903/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 854.
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.
\[ \boxed {y^{\prime }-\frac {y \left (\ln \left (x \right )+\ln \left (y\right )-1+\ln \left (x \right )^{2} x^{2}+2 x^{2} \ln \left (y\right ) \ln \left (x \right )+x^{2} \ln \left (y\right )^{2}\right )}{x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \ln \left (x \right )^{2} y x^{2}+2 y \ln \left (x \right ) \ln \left (y\right ) x^{2}+y \ln \left (y\right )^{2} x^{2}+y \ln \left (x \right )-y^{\prime } x +y \ln \left (y\right )-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-\ln \left (x \right )^{2} y x^{2}-2 y \ln \left (x \right ) \ln \left (y\right ) x^{2}-y \ln \left (y\right )^{2} x^{2}-y \ln \left (x \right )-y \ln \left (y\right )+y}{x} \end {array} \]
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(x)^2+2*ln(x)*ln(y)+ln(y)^2)/x]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 21
dsolve(diff(y(x),x) = y(x)*(ln(x)+ln(y(x))-1+x^2*ln(x)^2+2*x^2*ln(y(x))*ln(x)+x^2*ln(y(x))^2)/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {3 x}{x^{3}+3 c_{1}}}}{x} \]
✓ Solution by Mathematica
Time used: 0.349 (sec). Leaf size: 31
DSolve[y'[x] == ((-1 + Log[x] + x^2*Log[x]^2 + Log[y[x]] + 2*x^2*Log[x]*Log[y[x]] + x^2*Log[y[x]]^2)*y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {3 x}{x^3+3 c_1}}}{x} \\ y(x)\to \frac {1}{x} \\ \end{align*}