61.22.16 problem 16

Internal problem ID [12341]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 16
Date solved : Tuesday, January 28, 2025 at 07:54:55 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y-y&=\frac {A}{x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 57

dsolve(y(x)*diff(y(x),x)-y(x)=A*1/x,y(x), singsol=all)
 

\[ \frac {\operatorname {erf}\left (\frac {\left (y-x \right ) \sqrt {2}}{2 \sqrt {-A}}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 \,{\mathrm e}^{\frac {\left (y-x \right )^{2}}{2 A}} \sqrt {-A}+c_{1} x}{x} = 0 \]

✓ Solution by Mathematica

Time used: 0.508 (sec). Leaf size: 64

DSolve[y[x]*D[y[x],x]-y[x]==A*1/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\frac {x}{\sqrt {A}}=\frac {2 e^{\frac {(x-y(x))^2}{2 A}}}{\sqrt {2 \pi } \text {erfi}\left (\frac {y(x)-x}{\sqrt {2} \sqrt {A}}\right )+2 c_1},y(x)\right ] \]