79.3.3 problem 7 (iii)

Internal problem ID [18527]
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 : Tuesday, January 28, 2025 at 11:52:01 AM
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*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 47

dsolve([diff(x(t),t$2)+2*diff(x(t),t)+4*x(t)=exp(t)*cos(2*t),x(0) = 0, D(x)(0) = 1],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.180 (sec). Leaf size: 62

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

\[ x(t)\to \frac {1}{73} e^{-t} \left (8 e^{2 t} \sin (2 t)+17 \sqrt {3} \sin \left (\sqrt {3} t\right )+3 e^{2 t} \cos (2 t)-3 \cos \left (\sqrt {3} t\right )\right ) \]