32.4.11 problem 12

Internal problem ID [7784]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 25. Second order differential equations. Further problems 25. page 1094
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 05:05:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+3 x&={\mathrm e}^{-3 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&={\frac {1}{2}} \\ x^{\prime }\left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 13

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+3*x(t) = exp(-3*t); 
ic:=[x(0) = 1/2, D(x)(0) = -2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {{\mathrm e}^{-3 t} \left (t -1\right )}{2} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 17

ode=D[x[t],{t,2}]+4*D[x[t],t]+3*x[t]==Exp[-3*t]; 
ic={x[0]==1/2,Derivative[1][x][0 ]==-2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {1}{2} e^{-3 t} (t-1) \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(3*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) - exp(-3*t),0) 
ics = {x(0): 1/2, Subs(Derivative(x(t), t), t, 0): -2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {1}{2} - \frac {t}{2}\right ) e^{- 3 t} \]