76.15.12 problem 12

Internal problem ID [17582]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 12
Date solved : Monday, March 31, 2025 at 04:18:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

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

\[ y = \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 -\frac {5 \sin \left (2 t \right )}{9}-\frac {t \cos \left (2 t \right )}{3} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 33

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

\[ y(t)\to -\frac {5}{9} \sin (2 t)-\frac {1}{3} t \cos (2 t)+c_1 \cos (t)+c_2 \sin (t) \]

✓ Sympy. Time used: 0.114 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - \frac {t \cos {\left (2 t \right )}}{3} - \frac {5 \sin {\left (2 t \right )}}{9} \]