55.27.2 problem 2

Internal problem ID [13775]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2 Equations Containing Power Functions. page 213
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:06:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

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

\[ y = c_1 \operatorname {AiryAi}\left (\frac {a x +b}{\left (-a \right )^{{2}/{3}}}\right )+c_2 \operatorname {AiryBi}\left (\frac {a x +b}{\left (-a \right )^{{2}/{3}}}\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 36

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

\begin{align*} y(x)&\to c_1 \operatorname {AiryAi}\left (\frac {b+a x}{a^{2/3}}\right )+c_2 \operatorname {AiryBi}\left (\frac {b+a x}{a^{2/3}}\right ) \end{align*}

✓ Sympy. Time used: 0.045 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = C_{1} Ai\left (\sqrt [3]{a} x + \frac {b}{a^{\frac {2}{3}}}\right ) + C_{2} Bi\left (\sqrt [3]{a} x + \frac {b}{a^{\frac {2}{3}}}\right ) \]