85.49.5 problem 1 (e)

Internal problem ID [22806]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 194
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 09:14:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 27

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

\[ i = \sin \left (\frac {t}{2}\right ) c_2 +\cos \left (\frac {t}{2}\right ) c_1 +t^{2}-8-\frac {2 \cos \left (4 t \right )}{63} \]

✓ Mathematica. Time used: 0.235 (sec). Leaf size: 36

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

\begin{align*} i(t)&\to t^2-\frac {2}{63} \cos (4 t)+c_1 \cos \left (\frac {t}{2}\right )+c_2 \sin \left (\frac {t}{2}\right )-8 \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 29

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

\[ i{\left (t \right )} = C_{1} \sin {\left (\frac {t}{2} \right )} + C_{2} \cos {\left (\frac {t}{2} \right )} + t^{2} - \frac {2 \cos {\left (4 t \right )}}{63} - 8 \]