61.34.29 problem 29

Internal problem ID [12714]
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 : 29
Date solved : Monday, March 31, 2025 at 06:52:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 1.299 (sec). Leaf size: 332

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

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

✗ Sympy

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

False