7.20.7 problem 36

Internal problem ID [561]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.4 (Derivatives, Integrals and products of transforms). Problems at page 303
Problem number : 36
Date solved : Tuesday, March 04, 2025 at 11:26:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x&=f \left (t \right ) \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: 0.395 (sec). Leaf size: 23

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

\[ x \left (t \right ) = -\frac {\left (\int _{0}^{t}f \left (\textit {\_U1} \right ) \sin \left (-2 t +2 \textit {\_U1} \right )d \textit {\_U1} \right )}{2} \]

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 99

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

\[ x(t)\to -\sin (2 t) \int _1^0\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]+\sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]+\cos (2 t) \left (\int _1^t-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]-\int _1^0-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]\right ) \]

✓ Sympy. Time used: 1.290 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x = Function("x") 
f = Function("f") 
ode = Eq(-f(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)
 

\[ x{\left (t \right )} = \left (- \frac {\int f{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} f{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2}\right ) \cos {\left (2 t \right )} + \left (\frac {\int f{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} f{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} \]