75.33.8 problem 837

Internal problem ID [17290]
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 : Tuesday, January 28, 2025 at 09:58:57 AM
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*}

✓ Solution by Maple

Time used: 8.582 (sec). Leaf size: 10

dsolve([diff(x(t),t$2)=cos(t),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 57

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

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