55.28.2 problem 12

Internal problem ID [13785]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 12
Date solved : Thursday, October 02, 2025 at 08:06:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 49

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

\[ y = {\mathrm e}^{-\frac {a x}{2}} \left (c_1 \operatorname {AiryAi}\left (\frac {a^{2}-4 b x -4 c}{4 b^{{2}/{3}}}\right )+c_2 \operatorname {AiryBi}\left (\frac {a^{2}-4 b x -4 c}{4 b^{{2}/{3}}}\right )\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to e^{-\frac {a x}{2}} \left (c_1 \operatorname {AiryAi}\left (\frac {a^2-4 (c+b x)}{4 (-b)^{2/3}}\right )+c_2 \operatorname {AiryBi}\left (\frac {a^2-4 (c+b x)}{4 (-b)^{2/3}}\right )\right ) \end{align*}

✗ Sympy

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

False