61.34.23 problem 23

Internal problem ID [12708]
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 : 23
Date solved : Monday, March 31, 2025 at 06:51:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.824 (sec). Leaf size: 164

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

\[ y(x)\to (-1)^{-\frac {\lambda }{\mu }} 2^{\frac {\lambda +\mu }{2 \mu }} \left (\left (e^x\right )^{\lambda }\right )^{\frac {\lambda -1}{2 \lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\mu }{2}} e^{-\frac {a \left (e^x\right )^{\lambda }}{\lambda }} \left (\left (e^x\right )^{\mu }\right )^{\frac {\lambda +\mu }{2 \mu }} \left (-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}\right )^{-\frac {\lambda }{2 \mu }} \left (c_1 (-1)^{\lambda /\mu } \operatorname {BesselI}\left (\frac {\lambda }{\mu },2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )+c_2 K_{\frac {\lambda }{\mu }}\left (2 \sqrt {-\frac {c \left (e^x\right )^{\mu }}{\mu ^2}}\right )\right ) \]

✗ Sympy

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

False