44.14.13 problem 2(e)

Internal problem ID [9337]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 2(e)
Date solved : Tuesday, September 30, 2025 at 06:16:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (2\right )&=0 \\ y^{\prime }\left (2\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.144 (sec). Leaf size: 46

ode:=diff(diff(y(x),x),x)+y(x) = exp(-x); 
ic:=[y(2) = 0, D(y)(2) = -2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sin \left (x \right ) \left (\left ({\mathrm e}^{-2}-4\right ) \cos \left (2\right )-{\mathrm e}^{-2} \sin \left (2\right )\right )}{2}-\frac {\cos \left (x \right ) \left (\left (\sin \left (2\right )+\cos \left (2\right )\right ) {\mathrm e}^{-2}-4 \sin \left (2\right )\right )}{2}+\frac {{\mathrm e}^{-x}}{2} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 98

ode=D[y[x],{x,2}]+y[x]==Exp[-x]; 
ic={y[2]==0,Derivative[1][y][2]==-2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sin (x) \int _1^2e^{-K[2]} \cos (K[2])dK[2]+\sin (x) \int _1^xe^{-K[2]} \cos (K[2])dK[2]-\cos (x) \int _1^2-e^{-K[1]} \sin (K[1])dK[1]+\cos (x) \int _1^x-e^{-K[1]} \sin (K[1])dK[1]+2 \sin (2-x) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 87

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

\[ y{\left (x \right )} = \frac {\left (- \sin {\left (2 \right )} + \cos {\left (2 \right )} - 4 e^{2} \cos {\left (2 \right )}\right ) \sin {\left (x \right )}}{2 e^{2} \cos ^{2}{\left (2 \right )} + 2 e^{2} \sin ^{2}{\left (2 \right )}} + \frac {\left (- \sin {\left (2 \right )} - \cos {\left (2 \right )} + 4 e^{2} \sin {\left (2 \right )}\right ) \cos {\left (x \right )}}{2 e^{2} \cos ^{2}{\left (2 \right )} + 2 e^{2} \sin ^{2}{\left (2 \right )}} + \frac {e^{- x}}{2} \]