1.19.3 problem 29

Internal problem ID [543]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 04:01:10 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.100 (sec). Leaf size: 14

ode:=diff(diff(x(t),t),t)-4*x(t) = 3*t; 
ic:=[x(0) = 0, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = -\frac {3 t}{4}+\frac {3 \sinh \left (2 t \right )}{8} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 29

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

\begin{align*} x(t)&\to \frac {3}{16} e^{-2 t} \left (-4 e^{2 t} t+e^{4 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.059 (sec). Leaf size: 24

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

\[ x{\left (t \right )} = - \frac {3 t}{4} + \frac {3 e^{2 t}}{16} - \frac {3 e^{- 2 t}}{16} \]