61.34.27 problem 27

Internal problem ID [12712]
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 : 27
Date solved : Friday, March 14, 2025 at 12:17:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 1.237 (sec). Leaf size: 141

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

\[ y = {\mathrm e}^{\frac {-{\mathrm e}^{\lambda x} a -\lambda x \left (b +\lambda \right )}{2 \lambda }} \left (\operatorname {WhittakerM}\left (-\frac {\left (b +\lambda \right ) a -2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {\left (b +\lambda \right ) a -2 \beta }{2 \sqrt {a^{2}-4 \alpha }\, \lambda }, \frac {\sqrt {b^{2}-4 \gamma }}{2 \lambda }, \frac {\sqrt {a^{2}-4 \alpha }\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 1.812 (sec). Leaf size: 250

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {-2 \beta +a (b+\lambda )+\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda },\frac {\lambda +\sqrt {b^2-4 \gamma }}{\lambda },\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )+c_2 L_{\frac {2 \beta -a (b+\lambda )-\sqrt {a^2-4 \alpha } \left (\lambda +\sqrt {b^2-4 \gamma }\right )}{2 \sqrt {a^2-4 \alpha } \lambda }}^{\frac {\sqrt {b^2-4 \gamma }}{\lambda }}\left (\frac {\sqrt {a^2-4 \alpha } e^{x \lambda }}{\lambda }\right )\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {b+a K[1]+\sqrt {a^2-4 \alpha } K[1]-\sqrt {b^2-4 \gamma }}{2 \lambda K[1]}dK[1]\right ) \]

✗ Sympy

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

False