2.24.5 Problem 5

Internal problem ID [13569]
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 1.3.3-2.
Problem number : 5
Date solved : Friday, December 19, 2025 at 07:06:53 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }+x \left (a \,x^{2}+b \right ) y+x&=0 \\ \end{align*}

Unknown ode type.

2.24.5.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 179

ode:=y(x)*diff(y(x),x)+x*(a*x^2+b)*y(x)+x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+x \left (a \,x^{2}+b \right ) y \left (x \right )+x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {x \left (a \,x^{2}+b \right ) y \left (x \right )+x}{y \left (x \right )} \end {array} \]

2.24.5.2 ✓ Mathematica. Time used: 0.185 (sec). Leaf size: 143

ode=y[x]*D[y[x],x]+x*(a*x^2+b)*y[x]+x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

2.24.5.3 ✗ Sympy

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

NotImplementedError : The given ODE a*x**3 + b*x + x/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method