Internal problem ID [10727]
Internal file name [OUTPUT/9675_Monday_June_06_2022_03_21_35_PM_22910527/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 1.3.2. Equations of
the form \(y y'=f(x) y+1\)
Problem number: 3.
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.
\[ \boxed {y y^{\prime }-\left (a -\frac {1}{a x}\right ) y=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } a x -y a^{2} x -a x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y a^{2} x +a x -y}{a x y} \end {array} \]
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: 39
dsolve(y(x)*diff(y(x),x)=(a-1/(a*x))*y(x)+1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {a^{2} x -\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {expIntegral}_{1}\left (-\textit {\_Z} \right ) a^{2} x +c_{1} a^{2} x \right )}{a} \]
✓ Solution by Mathematica
Time used: 0.268 (sec). Leaf size: 37
DSolve[y[x]*y'[x]==(a-1/(a*x))*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\operatorname {ExpIntegralEi}(a (a x-y(x)))+c_1=\frac {e^{a (a x-y(x))}}{a^2 x},y(x)\right ] \]