Internal problem ID [10742]
Internal file name [OUTPUT/9690_Monday_June_06_2022_03_22_17_PM_23437655/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: 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }+a \left (1-\frac {1}{x}\right ) y=a^{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } x +a x y-a^{2} x -a y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-a x y+a^{2} x +a y}{x y} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(y(x)*diff(y(x),x)+a*(1-x^(-1))*y(x)=a^2,y(x), singsol=all)
\[ y \left (x \right ) = a \left (-x +\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right ) x +c_{1} x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.208 (sec). Leaf size: 30
DSolve[y[x]*y'[x]+a*(1-x^(-1))*y[x]==a^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\operatorname {ExpIntegralEi}\left (x+\frac {y(x)}{a}\right )+c_1=\frac {e^{\frac {y(x)}{a}+x}}{x},y(x)\right ] \]