2.23.1 Problem 1

Internal problem ID [13554]
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 1.3.2.
Problem number : 1
Date solved : Friday, December 19, 2025 at 06:50:44 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=\left (x a +b \right ) y+1 \\ \end{align*}

Unknown ode type.

2.23.1.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 215

ode:=y(x)*diff(y(x),x) = (a*x+b)*y(x)+1; 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {-\left (a x +b \right ) \left (-a^{2}\right )^{{1}/{3}} \left (-\operatorname {AiryBi}\left (-\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_1 +\operatorname {AiryAi}\left (-\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right ) 2^{{1}/{3}}+2 a \left (\operatorname {AiryBi}\left (1, -\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_1 -\operatorname {AiryAi}\left (1, -\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right )}{2^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} \left (a x +b \right ) \operatorname {AiryBi}\left (-\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{{2}/{3}} \left (-2 y a +\left (a x +b \right )^{2}\right )}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) a} = 0 \]

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 
<- Abel successful
 

Maple step by step

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

2.23.1.2 ✓ Mathematica. Time used: 0.461 (sec). Leaf size: 179

ode=y[x]*D[y[x],x]==(a*x+b)*y[x]+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {(-1)^{2/3} \sqrt [3]{2} (a x+b) \operatorname {AiryAi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left ((b+a x)^2-2 a y(x)\right )}{2 a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left ((b+a x)^2-2 a y(x)\right )}{2 a^{2/3}}\right )}{(-1)^{2/3} \sqrt [3]{2} (a x+b) \operatorname {AiryBi}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left ((b+a x)^2-2 a y(x)\right )}{2 a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (-\frac {\sqrt [3]{-\frac {1}{2}} \left ((b+a x)^2-2 a y(x)\right )}{2 a^{2/3}}\right )}+c_1=0,y(x)\right ] \]

2.23.1.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((-a*x - b)*y(x) + y(x)*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*x - b + Derivative(y(x), x) - 1/y(x) cannot be solved by the
 

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')