61.34.28 problem 28

Internal problem ID [12713]
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 : 28
Date solved : Wednesday, March 05, 2025 at 08:18:21 PM
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}+b \,{\mathrm e}^{2 \mu x}+c \,{\mathrm e}^{\mu x}+k \right ) y&=0 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 1.067 (sec). Leaf size: 290

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

\[ y(x)\to \left (\left (e^x\right )^{\lambda }\right )^{\frac {\lambda -1}{2 \lambda }} \left (e^x\right )^{\frac {1}{2}-\frac {\mu }{2}} 2^{\frac {\sqrt {\mu ^2 \left (\lambda ^2-4 k\right )}+\mu ^2}{2 \mu ^2}} \left (\left (e^x\right )^{\mu }\right )^{\frac {\sqrt {\mu ^2 \left (\lambda ^2-4 k\right )}+\mu ^2}{2 \mu ^2}} e^{-\frac {a \left (e^x\right )^{\lambda }}{\lambda }+\frac {i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\mu ^2-\frac {i c \mu }{\sqrt {b}}+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{2 \mu ^2},\frac {\mu ^2+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{\mu ^2},-\frac {2 i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }\right )+c_2 L_{\frac {i c}{2 \sqrt {b} \mu }-\frac {\mu ^2+\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{2 \mu ^2}}^{\frac {\sqrt {\left (\lambda ^2-4 k\right ) \mu ^2}}{\mu ^2}}\left (-\frac {2 i \sqrt {b} \left (e^x\right )^{\mu }}{\mu }\right )\right ) \]

✗ Sympy

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

False