61.34.5 problem 5

Internal problem ID [12690]
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 : 5
Date solved : Monday, March 31, 2025 at 06:51:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.165 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 0.565 (sec). Leaf size: 152

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

\[ y(x)\to \left (e^{\lambda x}\right )^{\frac {\sqrt {c}}{\lambda }} e^{\frac {\sqrt {c}-\sqrt {a} e^{\lambda x}}{\lambda }} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\frac {b}{\sqrt {a}}+\lambda +2 \sqrt {c}}{2 \lambda },\frac {2 \sqrt {c}}{\lambda }+1,\frac {2 \sqrt {a} e^{x \lambda }}{\lambda }\right )+c_2 L_{-\frac {\frac {b}{\sqrt {a}}+\lambda +2 \sqrt {c}}{2 \lambda }}^{\frac {2 \sqrt {c}}{\lambda }}\left (\frac {2 \sqrt {a} e^{x \lambda }}{\lambda }\right )\right ) \]

✗ Sympy

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

False