44.14.11 problem 2(c)

Internal problem ID [9335]
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(c)
Date solved : Tuesday, September 30, 2025 at 06:16:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&={\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+5*y(x) = exp(x); 
ic:=[y(0) = -1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (-\sin \left (2 x \right )-9 \cos \left (2 x \right )\right ) {\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{x}}{8} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 118

ode=D[y[x],{x,2}]+2*D[y[x],x]+5*y[x]==Exp[x]; 
ic={y[0]==-1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (-\sin (2 x) \int _1^0\frac {1}{2} e^{2 K[1]} \cos (2 K[1])dK[1]+\sin (2 x) \int _1^x\frac {1}{2} e^{2 K[1]} \cos (2 K[1])dK[1]+\cos (2 x) \left (\int _1^x-e^{2 K[2]} \cos (K[2]) \sin (K[2])dK[2]-\int _1^0-e^{2 K[2]} \cos (K[2]) \sin (K[2])dK[2]-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - exp(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {\sin {\left (2 x \right )}}{8} - \frac {9 \cos {\left (2 x \right )}}{8}\right ) e^{- x} + \frac {e^{x}}{8} \]