14.11.16 problem 16

Internal problem ID [2609]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 16
Date solved : Sunday, March 30, 2025 at 12:11:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=\cos \left (t \right ) \cos \left (2 t \right ) \cos \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 32

ode:=diff(diff(y(t),t),t)+y(t) = cos(t)*cos(2*t)*cos(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {8 \cos \left (t \right )^{6}}{35}+\frac {22 \cos \left (t \right )^{4}}{105}+\cos \left (t \right ) c_1 +\sin \left (t \right ) c_2 -\frac {17 \cos \left (t \right )^{2}}{105}+\frac {34}{105} \]

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 43

ode=D[y[t],{t,2}]+y[t]==Cos[t]*Cos[2*t]*Cos[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {1}{12} \cos (2 t)-\frac {1}{60} \cos (4 t)-\frac {1}{140} \cos (6 t)+c_1 \cos (t)+c_2 \sin (t)+\frac {1}{4} \]

✓ Sympy. Time used: 3.034 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - cos(t)*cos(2*t)*cos(3*t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - \frac {3 \left (1 - \cos {\left (2 t \right )}\right )^{2} \cos {\left (2 t \right )}}{28} + \frac {3 \left (1 - \cos {\left (2 t \right )}\right )^{2} \cos {\left (4 t \right )}}{28} + \frac {71 \cos {\left (2 t \right )}}{336} - \frac {239 \cos {\left (4 t \right )}}{840} + \frac {71 \cos {\left (6 t \right )}}{560} - \frac {3 \cos {\left (8 t \right )}}{112} + \frac {13}{112} \]