2.22.14 Problem 17

Internal problem ID [13509]
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 : 17
Date solved : Friday, December 19, 2025 at 05:15:58 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-y&=-\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \\ \end{align*}

Unknown ode type.

2.22.14.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 209

ode:=y(x)*diff(y(x),x)-y(x) = -1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2/x^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +\frac {\left (-\sqrt {x}+3 A \right ) \sqrt {\frac {3 A^{2}+2 A \sqrt {x}-x +2 y}{6 A^{2}-2 A \sqrt {x}+y}}\, y \,{\mathrm e}^{\frac {-6 A^{2}+2 A \sqrt {x}-y}{3 A^{2}+2 A \sqrt {x}-x +2 y}}}{\sqrt {-\frac {\left (-\sqrt {x}+3 A \right )^{2}}{6 A^{2}-2 A \sqrt {x}+y}}\, \left (12 A^{3}-4 A^{2} \sqrt {x}+2 y A \right )}-\int _{}^{\frac {6 A \sqrt {x}-2 x +3 y}{12 A^{2}-4 A \sqrt {x}+2 y}}\frac {{\mathrm e}^{-\frac {2}{2 \textit {\_a} +1}} \sqrt {2 \textit {\_a} +1}}{\sqrt {2 \textit {\_a} -3}}d \textit {\_a} = 0 \]

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 
   Looking for potential symmetries 
   found: 2 potential symmetries. Proceeding with integration step 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=-\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )-\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}}{y \left (x \right )} \end {array} \]

2.22.14.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.14.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
y = Function("y") 
ode = Eq(-A*(3*A**2/sqrt(x) + 5*A + sqrt(x))/4 + x/4 + y(x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (3*A**3 + sqrt(x)*(5*A**2 + A*sqrt(x) - x) + 4*sqrt(x)*y(x))/(4*sqrt(x)*y(x)) cannot be solved by the factorable group method