61.34.4 problem 4

Internal problem ID [12689]
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 : 4
Date solved : Monday, March 31, 2025 at 06:51:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.151 (sec). Leaf size: 57

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

\[ y = \operatorname {WhittakerM}\left (-b , -b +\frac {1}{2}, 2 a \,{\mathrm e}^{x}\right ) c_2 +c_1 \,{\mathrm e}^{b x +a \,{\mathrm e}^{x}}-2 c_2 \left (a \,{\mathrm e}^{x}\right )^{-b} {\mathrm e}^{-a \,{\mathrm e}^{x}} \left (b -\frac {1}{2}\right ) 2^{-b} \]

✓ Mathematica. Time used: 1.038 (sec). Leaf size: 59

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

\[ y(x)\to e^{a e^x} \left (e^x\right )^b \left (c_2 \int _1^{e^x}\exp \left (\int _1^{K[2]}-\frac {2 b+2 a K[1]+1}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]

✗ Sympy

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

False