23.3.52 problem 54

Internal problem ID [5766]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 54
Date solved : Friday, October 03, 2025 at 01:43:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.122 (sec). Leaf size: 58

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

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

✓ Mathematica. Time used: 0.334 (sec). Leaf size: 136

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

\begin{align*} y(x)&\to \left (e^x\right )^{i \sqrt {a}} e^{-i \sqrt {c} e^x} \left (c_1 \operatorname {HypergeometricU}\left (\frac {i b}{2 \sqrt {c}}+i \sqrt {a}+\frac {1}{2},2 i \sqrt {a}+1,2 i \sqrt {c} e^x\right )+c_2 L_{-\frac {i b}{2 \sqrt {c}}-i \sqrt {a}-\frac {1}{2}}^{2 i \sqrt {a}}\left (2 i \sqrt {c} e^x\right )\right ) \end{align*}

✗ Sympy

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

False