61.24.5 problem 5

Internal problem ID [12339]
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, March 14, 2025 at 04:43:58 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*}

✓ Maple. Time used: 0.003 (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);
 

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

✓ Mathematica. Time used: 0.314 (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]
 

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

✗ 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