54.3.20 problem 1020

Internal problem ID [12315]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1020
Date solved : Friday, October 03, 2025 at 03:18:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.156 (sec). Leaf size: 52

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

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

✓ Mathematica. Time used: 0.362 (sec). Leaf size: 144

ode=(c + b*E^x + a*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 {c}} e^{i \left (\sqrt {c}-\sqrt {a} e^x\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {i b}{2 \sqrt {a}}+i \sqrt {c}+\frac {1}{2},2 i \sqrt {c}+1,2 i \sqrt {a} e^x\right )+c_2 L_{-\frac {i b}{2 \sqrt {a}}-i \sqrt {c}-\frac {1}{2}}^{2 i \sqrt {c}}\left (2 i \sqrt {a} 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*exp(2*x) + b*exp(x) + c)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False