2.22.31 Problem 36

Internal problem ID [13526]
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 : 36
Date solved : Friday, December 19, 2025 at 05:50:26 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.22.31.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 407

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

\[ \frac {-c_1 \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right )+A \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\, c_1 +\operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\, A +\operatorname {BesselK}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right ) \left (\sqrt {\frac {A^{3}-B}{A^{3}}}\, A -A -\sqrt {x}\right )}{\left (-\sqrt {\frac {A^{3}-B}{A^{3}}}\, A +A +\sqrt {x}\right ) \operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right )+\operatorname {BesselI}\left (\sqrt {\frac {A^{3}-B}{A^{3}}}+1, -\sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\right ) \sqrt {\frac {2 A^{2} \sqrt {x}-A y+A x +B}{A^{3}}}\, A} = 0 \]

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 
   Looking for potential symmetries 
<- Abel successful
 

Maple step by step

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

2.22.31.2 ✗ Mathematica

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

Not solved

2.22.31.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
B = symbols("B") 
y = Function("y") 
ode = Eq(-2*A**2 - A*sqrt(x) - B/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/y(x) - A*sqrt(x)/y(x) - B/(sqrt(x)*y(x)) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method