2.22.30 Problem 35

Internal problem ID [13525]
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 : 35
Date solved : Friday, December 19, 2025 at 05:46:14 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.22.30.1 ✓ Maple. Time used: 0.005 (sec). Leaf size: 474

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

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

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=13527 A \left (\sqrt {x}+27054 A +\frac {27053 A^{2}}{\sqrt {x}}\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+13527 A \left (\sqrt {x}+27054 A +\frac {27053 A^{2}}{\sqrt {x}}\right )}{y \left (x \right )} \end {array} \]

2.22.30.2 ✗ Mathematica

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

Not solved

2.22.30.3 ✗ Sympy

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

Timed Out
 

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