73.3.5 problem 7 (v)

Internal problem ID [19815]
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 : Thursday, October 02, 2025 at 04:44:05 PM
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*}

✓ Maple. Time used: 0.035 (sec). Leaf size: 35

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+3*x(t) = sin(t)*t; 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 42

ode=D[x[t],{t,2}]+4*D[x[t],t]+3*x[t]==t*Sin[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}{100} \left (e^{-3 t} \left (25 e^{2 t}-47\right )+2 (5 t+2) \sin (t)+(22-20 t) \cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*sin(t) + 3*x(t) + 4*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)
 

\[ x{\left (t \right )} = \frac {t \sin {\left (t \right )}}{10} - \frac {t \cos {\left (t \right )}}{5} + \frac {\sin {\left (t \right )}}{25} + \frac {11 \cos {\left (t \right )}}{50} + \frac {e^{- t}}{4} - \frac {47 e^{- 3 t}}{100} \]