83.21.3 problem 3

Internal problem ID [22063]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination VIII. page 256
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:23:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t)+y(t) = 2*cos(t); 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \cos \left (t \right )+\sin \left (t \right ) \left (1+t \right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 16

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

\begin{align*} y(t)&\to (t+1) \sin (t)+2 \cos (t) \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 14

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): 2, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t + 1\right ) \sin {\left (t \right )} + 2 \cos {\left (t \right )} \]