55.35.22 problem 22

Internal problem ID [14059]
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 : 22
Date solved : Friday, October 03, 2025 at 07:24:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.388 (sec). Leaf size: 205

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

\[ y = {\mathrm e}^{-\frac {{\mathrm e}^{\lambda x} \left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right )}{2 \lambda }} \left (\operatorname {KummerU}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (a +b -2 \lambda \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {a +b -2 \lambda }{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (b +\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\right ) \left (a +b -2 \lambda \right )}{2 \sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, \lambda }, \frac {a +b -2 \lambda }{\lambda }, \frac {\sqrt {-4 \lambda \left (b -\lambda \right ) a^{2}+b^{2}}\, {\mathrm e}^{\lambda x}}{\lambda }\right ) c_1 \right ) \]

✓ Mathematica. Time used: 1.227 (sec). Leaf size: 260

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

\begin{align*} y(x)&\to \exp \left (-\frac {e^{\lambda x} \left (\sqrt {-4 a^2 b \lambda +4 a^2 \lambda ^2+b^2}+b\right )}{2 \lambda }\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}},\frac {a+b-2 \lambda }{\lambda },\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )+c_2 L_{-\frac {(a+b-2 \lambda ) \left (b+\sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}\right )}{2 \lambda \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}}^{\frac {a+b-3 \lambda }{\lambda }}\left (\frac {e^{x \lambda } \sqrt {4 \lambda ^2 a^2-4 b \lambda a^2+b^2}}{\lambda }\right )\right ) \end{align*}

✗ Sympy

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

False