79.3.5 problem 7 (v)

Internal problem ID [18529]
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 (v)
Date solved : Tuesday, January 28, 2025 at 11:54:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+3 x&=t \sin \left (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: 0.020 (sec). Leaf size: 35

dsolve([diff(x(t),t$2)+4*diff(x(t),t)+3*x(t)=t*sin(t),x(0) = 0, D(x)(0) = 1],x(t), singsol=all)
 

\[ x = -\frac {47 \,{\mathrm e}^{-3 t}}{100}+\frac {{\mathrm e}^{-t}}{4}+\frac {\left (-10 t +11\right ) \cos \left (t \right )}{50}+\frac {\sin \left (t \right ) \left (5 t +2\right )}{50} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 42

DSolve[{D[x[t],{t,2}]+4*D[x[t],t]+3*x[t]==t*Sin[t],{x[0]==0,Derivative[1][x][0] == 1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{100} \left (e^{-3 t} \left (25 e^{2 t}-47\right )+2 (5 t+2) \sin (t)+(22-20 t) \cos (t)\right ) \]