69.33.15 problem 844

Internal problem ID [18584]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3. Section 24.2. Solving the Cauchy problem for linear differential equation with constant coefficients. Exercises page 249
Problem number : 844
Date solved : Thursday, October 02, 2025 at 03:15:06 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+4 x&=4 \end{align*}

Using Laplace method With initial conditions

\begin{align*} x \left (0\right )&=1 \\ x^{\prime }\left (0\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.096 (sec). Leaf size: 13

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+4*x(t) = 4; 
ic:=[x(0) = 1, D(x)(0) = -4]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = -4 \,{\mathrm e}^{-2 t} t +1 \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 15

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

\begin{align*} x(t)&\to 1-4 e^{-2 t} t \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 12

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

\[ x{\left (t \right )} = - 4 t e^{- 2 t} + 1 \]