68.18.39 problem 45

Internal problem ID [17883]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 45
Date solved : Thursday, October 02, 2025 at 02:29:12 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\cos \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 9

ode:=diff(diff(y(t),t),t)+y(t) = cos(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\sin \left (t \right ) t}{2} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 44

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

\begin{align*} y(t)&\to -\frac {1}{4} \sin (t) \left (-4 \int _1^t\cos ^2(K[1])dK[1]+4 \int _1^0\cos ^2(K[1])dK[1]+\sin (2 t)\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 8

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

\[ y{\left (t \right )} = \frac {t \sin {\left (t \right )}}{2} \]