2.22.32 Problem 37

Internal problem ID [13527]
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 : 37
Date solved : Friday, December 19, 2025 at 05:52:52 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }-y&=2 A^{2}-A \sqrt {x} \\ \end{align*}

Unknown ode type.

2.22.32.1 ✓ Maple. Time used: 0.001 (sec). Leaf size: 228

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

\[ \frac {\left (-2 A +\sqrt {x}\right ) \operatorname {BesselK}\left (1, -\sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )+\operatorname {BesselK}\left (0, -\sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right ) \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\, A +c_1 \left (A \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\, \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )\right )}{A \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\, \operatorname {BesselI}\left (0, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )+\left (-2 A +\sqrt {x}\right ) \operatorname {BesselI}\left (1, \sqrt {-\frac {2 A \sqrt {x}-x +y}{A^{2}}}\right )} = 0 \]

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 
   Looking for potential symmetries 
   Looking for potential symmetries 
<- Abel successful
 

Maple step by step

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

2.22.32.2 ✗ Mathematica

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

Not solved

2.22.32.3 ✗ Sympy

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

NotImplementedError : The given ODE -(2*A**2 - A*sqrt(x) + y(x))/y(x) + Derivative(y(x), x) cannot b
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '1st_power_series', 'lie_group')