61.34.16 problem 16

Internal problem ID [12701]
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 : 16
Date solved : Wednesday, March 05, 2025 at 08:17:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 1.527 (sec). Leaf size: 232

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

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

✗ 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*k*exp(mu*x)*Derivative(y(x), x) + (a*exp(2*cg*x) + b*exp(cg*x) + c + k**2*exp(2*mu*x) + k*mu*exp(mu*x))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False