Internal problem ID [10743]
Internal file name [OUTPUT/9691_Monday_June_06_2022_03_22_18_PM_56791468/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: 7.
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.
\[ \boxed {y y^{\prime }-a \left (1-\frac {b}{x}\right ) y=a^{2} b} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -a^{2} b x +y y^{\prime } x +a y b -a x y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a^{2} b x -a y b +a x y}{x y} \end {array} \]
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: 29
dsolve(y(x)*diff(y(x),x)-a*(1-b*x^(-1))*y(x)=a^2*b,y(x), singsol=all)
\[ y \left (x \right ) = a \left (-\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} b -\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right ) x +c_{1} x \right ) b +x \right ) \]
✓ Solution by Mathematica
Time used: 0.293 (sec). Leaf size: 45
DSolve[y[x]*y'[x]-a*(1-b*x^(-1))*y[x]==a^2*b,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\operatorname {ExpIntegralEi}\left (\frac {a x-y(x)}{a b}\right )+c_1=\frac {b e^{\frac {a x-y(x)}{a b}}}{x},y(x)\right ] \]