Internal problem ID [9087]
Internal file name [OUTPUT/8022_Monday_June_06_2022_01_19_01_AM_27223529/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 753.
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.
\[ \boxed {y^{\prime }-\frac {\left (x +1+x^{4} \ln \left (y\right )\right ) y \ln \left (y\right )}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \ln \left (y\right )^{2} y x^{4}+\ln \left (y\right ) y x -y^{\prime } x^{2}+y \ln \left (y\right )-y^{\prime } x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\ln \left (y\right )^{2} y x^{4}+\ln \left (y\right ) y x +y \ln \left (y\right )}{-x^{2}-x} \end {array} \]
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: 0.016 (sec). Leaf size: 38
dsolve(diff(y(x),x) = (x+1+x^4*ln(y(x)))*y(x)*ln(y(x))/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {12 x}{3 x^{4}-4 x^{3}+6 x^{2}+12 \ln \left (x +1\right )-12 c_{1} -12 x}} \]
✓ Solution by Mathematica
Time used: 0.5 (sec). Leaf size: 46
DSolve[y'[x] == (Log[y[x]]*(1 + x + x^4*Log[y[x]])*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (\frac {12 x}{-3 x^4+4 x^3-6 x^2+12 x-12 \log (x+1)+12 c_1}\right ) \\ y(x)\to 1 \\ \end{align*}