61.34.11 problem 11

Internal problem ID [12696]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.3-1. Equations with exponential functions
Problem number : 11
Date solved : Friday, March 14, 2025 at 12:16:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y^{\prime }+\left (a \,{\mathrm e}^{3 \lambda x}+b \,{\mathrm e}^{2 \lambda x}+\frac {1}{4}-\frac {\lambda ^{2}}{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.790 (sec). Leaf size: 51

ode:=diff(diff(y(x),x),x)-diff(y(x),x)+(a*exp(3*lambda*x)+b*exp(2*lambda*x)+1/4-1/4*lambda^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.634 (sec). Leaf size: 77

ode=D[y[x],{x,2}]-D[y[x],x]+(a*Exp[3*\[Lambda]*x]+b*Exp[2*\[Lambda]*x]+1/4-1/4*\[Lambda]^2  )*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
cg = symbols("cg") 
y = Function("y") 
ode = Eq((a*exp(3*cg*x) + b*exp(2*cg*x) - cg**2/4 + 1/4)*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False