Internal problem ID [10664]
Internal file name [OUTPUT/9612_Monday_June_06_2022_03_15_48_PM_62995207/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 16.
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.
\[ \boxed {y y^{\prime }-y=\frac {A}{x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } x +x y+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x y+A}{x y} \end {array} \]
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: 57
dsolve(y(x)*diff(y(x),x)-y(x)=A*1/x,y(x), singsol=all)
\[ \frac {\operatorname {erf}\left (\frac {\left (y \left (x \right )-x \right ) \sqrt {2}}{2 \sqrt {-A}}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 \,{\mathrm e}^{\frac {\left (y \left (x \right )-x \right )^{2}}{2 A}} \sqrt {-A}+c_{1} x}{x} = 0 \]
✓ Solution by Mathematica
Time used: 0.827 (sec). Leaf size: 64
DSolve[y[x]*y'[x]-y[x]==A*1/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {x}{\sqrt {A}}=\frac {2 e^{\frac {(x-y(x))^2}{2 A}}}{\sqrt {2 \pi } \text {erfi}\left (\frac {y(x)-x}{\sqrt {2} \sqrt {A}}\right )+2 c_1},y(x)\right ] \]