61.34.26 problem 26

Internal problem ID [12711]
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 : 26
Date solved : Monday, March 31, 2025 at 06:52:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.118 (sec). Leaf size: 89

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

\[ y = c_1 \,{\mathrm e}^{-k x -{\mathrm e}^{x} c}+c_2 \,{\mathrm e}^{-\frac {a \,{\mathrm e}^{x}}{2}-\frac {b x}{2}} \left (-2 c +a \right ) \left (-{\mathrm e}^{-\frac {\left (-2 c +a \right ) {\mathrm e}^{x}}{2}} \left (-1-2 k +b \right ) \left (\left (-2 c +a \right ) {\mathrm e}^{x}\right )^{k -\frac {b}{2}}+\operatorname {WhittakerM}\left (k -\frac {b}{2}, k -\frac {b}{2}+\frac {1}{2}, \left (-2 c +a \right ) {\mathrm e}^{x}\right )\right ) \]

✓ Mathematica. Time used: 1.931 (sec). Leaf size: 71

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

\[ y(x)\to \left (e^x\right )^{-k} e^{c \left (-e^x\right )-k} \left (c_2 \int _1^{e^x}\exp \left (\int _1^{K[2]}-\frac {b-2 k+a K[1]-2 c K[1]+1}{K[1]}dK[1]\right )dK[2]+c_1\right ) \]

✗ Sympy

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

False