69.33.8 problem 837

Internal problem ID [18577]
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 : 837
Date solved : Thursday, October 02, 2025 at 03:15:03 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} x^{\prime \prime }&=\cos \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: 0.108 (sec). Leaf size: 10

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

\[ x = -\cos \left (t \right )+1 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 57

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

\begin{align*} x(t)&\to -t \int _1^0\cos (K[1])dK[1]+\int _1^t\int _1^{K[2]}\cos (K[1])dK[1]dK[2]-\int _1^0\int _1^{K[2]}\cos (K[1])dK[1]dK[2] \end{align*}

✓ Sympy. Time used: 0.033 (sec). Leaf size: 7

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

\[ x{\left (t \right )} = 1 - \cos {\left (t \right )} \]