Internal problem ID [10813]
Internal file name [OUTPUT/9761_Thursday_June_09_2022_12_57_50_AM_87747449/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 77.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (2 \ln \left (x \right )+a +1\right ) y=x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (2 \ln \left (x \right )+a +1\right ) y=x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (2 \ln \left (x \right )+a +1\right ) y+x \left (-\ln \left (x \right )^{2}-a \ln \left (x \right )+b \right )}{y} \end {array} \]
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 trying Abel Looking for potential symmetries found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 163
dsolve(y(x)*diff(y(x),x)=(2*ln(x)+a+1)*y(x)+x*( -(ln(x))^2-a*ln(x)+b),y(x), singsol=all)
\[ y \left (x \right ) = \frac {x \left (-\tanh \left (\frac {\operatorname {RootOf}\left (-\sqrt {a^{2}+4 b}\, \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} \tanh \left (\frac {\textit {\_Z} \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}} a -{\mathrm e}^{-\frac {2 \,\operatorname {arctanh}\left (\frac {2 \textit {\_a} -a}{\sqrt {a^{2}+4 b}}\right )}{\sqrt {a^{2}+4 b}}} a +2 c_{1} \right ) \sqrt {a^{2}+4 b}}{2}\right ) \sqrt {a^{2}+4 b}+2 \ln \left (x \right )+a \right )}{2} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(2*Log[x]+a+1)*y[x]+x*( -(Log[x])^2-a*Log[x]+b),y[x],x,IncludeSingularSolutions -> True]
Not solved