74.9.9 problem 17

Internal problem ID [16114]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 17
Date solved : Monday, March 31, 2025 at 02:44:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=2 \cos \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 13

ode:=diff(diff(y(t),t),t)+y(t) = 2*cos(t); 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \sin \left (t \right )+\cos \left (t \right )+\sin \left (t \right ) t \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 47

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

\[ y(t)\to -\sin (t) \int _1^02 \cos ^2(K[1])dK[1]+\sin (t) \int _1^t2 \cos ^2(K[1])dK[1]+\sin (t)+\cos ^3(t) \]

✓ Sympy. Time used: 0.090 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 2*cos(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t + 1\right ) \sin {\left (t \right )} + \cos {\left (t \right )} \]