2.22.15 Problem 18

Internal problem ID [13510]
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 : 18
Date solved : Friday, December 19, 2025 at 05:18:25 AM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \\ \end{align*}

Unknown ode type.

2.22.15.1 ✓ Maple. Time used: 0.006 (sec). Leaf size: 710

ode:=y(x)*diff(y(x),x)-y(x) = 2*a^2/(8*a^2+x^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {512 \left (-\frac {33 \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) x \sqrt {8 a^{2}+x^{2}}}{64}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right ) a \,{\mathrm e}^{-\frac {\left (x -y\right )^{2} \left (-64 \sqrt {8 a^{2}+x^{2}}\, a^{6}-108 \sqrt {8 a^{2}+x^{2}}\, a^{4} x^{2}-25 a^{2} \sqrt {8 a^{2}+x^{2}}\, x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{6}+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )^{2}}{2 \left (128 a^{6}+150 a^{4} x^{2}-66 \sqrt {8 a^{2}+x^{2}}\, x \,a^{4}+27 a^{2} x^{4}-23 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{3}+x^{6}-\sqrt {8 a^{2}+x^{2}}\, x^{5}\right )^{2} a^{2} \left (-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}\right )}}+128 \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \left (x -y\right ) \left (\sqrt {8 a^{2}+x^{2}}\, \left (-64 a^{6}-108 a^{4} x^{2}-25 a^{2} x^{4}-x^{6}\right )+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )}{2 \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-66 a^{5} x -23 a^{3} x^{3}-a \,x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+128 a^{7}+150 a^{5} x^{2}+27 a^{3} x^{4}+a \,x^{6}\right )}\right ) \sqrt {2}\, \sqrt {\pi }+c_1 \right ) \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\frac {\left (\left (a^{4}+\frac {21}{32} a^{2} x^{2}+\frac {1}{32} x^{4}\right ) y-\frac {33 \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) x}{16}\right ) \sqrt {8 a^{2}+x^{2}}}{4}+\frac {\left (-25 a^{4} x -\frac {25}{4} a^{2} x^{3}-\frac {1}{4} x^{5}\right ) y}{32}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right )}{\sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (\left (32 a^{4}+21 a^{2} x^{2}+x^{4}\right ) y-66 a^{4} x -23 a^{2} x^{3}-x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+\left (-100 a^{4} x -25 a^{2} x^{3}-x^{5}\right ) y+128 a^{6}+150 a^{4} x^{2}+27 a^{2} x^{4}+x^{6}\right )} = 0 \]

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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}}{y \left (x \right )} \end {array} \]

2.22.15.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==2*a^2/Sqrt[x^2+8*a^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.15.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a**2/sqrt(8*a**2 + x**2) + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -2*a**2/(sqrt(8*a**2 + x**2)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method