68.11.25 problem 37

Internal problem ID [17562]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 37
Date solved : Thursday, October 02, 2025 at 02:25:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=-32 t^{2} \cos \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 35

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = -32*t^2*cos(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-4 t +3\right ) \cos \left (2 t \right )+\left (c_1 t +c_2 \right ) {\mathrm e}^{-2 t}-4 \sin \left (2 t \right ) t \left (t -1\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 44

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==-32*t^2*Cos[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to (3-4 t) \cos (2 t)+e^{-2 t} \left (-4 e^{2 t} (t-1) t \sin (2 t)+c_2 t+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.194 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(32*t**2*cos(2*t) + 4*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 4 t^{2} \sin {\left (2 t \right )} + 4 t \sin {\left (2 t \right )} - 4 t \cos {\left (2 t \right )} + \left (C_{1} + C_{2} t\right ) e^{- 2 t} + 3 \cos {\left (2 t \right )} \]