78.16.5 problem 5

Internal problem ID [18313]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 23. Operator Methods for Finding Particular Solutions. Problems at page 169
Problem number : 5
Date solved : Monday, March 31, 2025 at 05:25:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-y&={\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)-y(x) = exp(-x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 32

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

\[ y(x)\to \frac {1}{4} e^{-x} \left (-2 x+4 c_1 e^{2 x}-1+4 c_2\right ) \]

✓ Sympy. Time used: 0.086 (sec). Leaf size: 15

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

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