73.3.3 problem 7 (iii)

Internal problem ID [19813]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 5. Linear equations. Exercises at page 85
Problem number : 7 (iii)
Date solved : Thursday, October 02, 2025 at 04:44:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }+4 x&={\mathrm e}^{t} \cos \left (2 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.064 (sec). Leaf size: 47

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)+4*x(t) = exp(t)*cos(2*t); 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {3 \,{\mathrm e}^{-t} \cos \left (\sqrt {3}\, t \right )}{73}+\frac {17 \,{\mathrm e}^{-t} \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{73}+\frac {3 \,{\mathrm e}^{t} \left (\cos \left (2 t \right )+\frac {8 \sin \left (2 t \right )}{3}\right )}{73} \]

✓ Mathematica. Time used: 0.755 (sec). Leaf size: 62

ode=D[x[t],{t,2}]+2*D[x[t],t]+4*x[t]==Exp[t]*Cos[2*t]; 
ic={x[0]==0,Derivative[1][x][0] == 1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{73} e^{-t} \left (8 e^{2 t} \sin (2 t)+17 \sqrt {3} \sin \left (\sqrt {3} t\right )+3 e^{2 t} \cos (2 t)-3 \cos \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.222 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) - exp(t)*cos(2*t) + 2*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {17 \sqrt {3} \sin {\left (\sqrt {3} t \right )}}{73} - \frac {3 \cos {\left (\sqrt {3} t \right )}}{73}\right ) e^{- t} + \frac {\left (8 \sin {\left (2 t \right )} + 3 \cos {\left (2 t \right )}\right ) e^{t}}{73} \]