81.13.11 problem 17-11

Internal problem ID [21685]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 17. Reduction of Order. Page 420
Problem number : 17-11
Date solved : Friday, October 03, 2025 at 07:59:00 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 s y^{\prime }-2 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{s^{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)-2*s*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{2 x \sqrt {s^{2}+2}}+c_2 \right ) {\mathrm e}^{\left (s -\sqrt {s^{2}+2}\right ) x} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 44

ode=D[y[x],{x,2}]-2*s*D[y[x],x]-2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{\left (s-\sqrt {s^2+2}\right ) x}+c_2 e^{\left (\sqrt {s^2+2}+s\right ) x} \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} e^{x \left (s - \sqrt {s^{2} + 2}\right )} + C_{2} e^{x \left (s + \sqrt {s^{2} + 2}\right )} \]