2.23.2 Problem 2

Internal problem ID [13555]
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 : 2
Date solved : Friday, December 19, 2025 at 06:51:19 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }&=\frac {y}{\left (x a +b \right )^{2}}+1 \\ \end{align*}

Unknown ode type.

2.23.2.1 ✓ Maple. Time used: 0.003 (sec). Leaf size: 557

ode:=y(x)*diff(y(x),x) = 1/(a*x+b)^2*y(x)+1; 
dsolve(ode,y(x), singsol=all);
 

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

Maple step by step

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

2.23.2.2 ✓ Mathematica. Time used: 0.803 (sec). Leaf size: 561

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

\[ \text {Solve}\left [\frac {a y(x) (a x+b) \operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryAiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryBiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ] \]

2.23.2.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), x) - 1 - y(x)/(a*x + b)**2,0) 
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