68.12.10 problem 10

Internal problem ID [17604]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 10
Date solved : Thursday, October 02, 2025 at 02:26:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+25 y&={\mathrm e}^{-3 t} \left (\sec \left (4 t \right )+\csc \left (4 t \right )\right ) \end{align*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 54

ode:=diff(diff(y(t),t),t)+6*diff(y(t),t)+25*y(t) = exp(-3*t)*(sec(4*t)+csc(4*t)); 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 52

ode=D[y[t],{t,2}]+6*D[y[t],t]+25*y[t]==Exp[-3*t]*(Sec[4*t]+Csc[4*t]); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{16} e^{-3 t} (\cos (4 t) (-4 t+\log (\cos (4 t))+16 c_2)+\sin (4 t) (\log (\sin (4 t))+4 (t+4 c_1))) \end{align*}

✓ Sympy. Time used: 1.047 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-1/cos(4*t) - 1/sin(4*t))*exp(-3*t) + 25*y(t) + 6*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - \frac {t}{4} + \frac {\log {\left (\cos {\left (4 t \right )} \right )}}{16}\right ) \cos {\left (4 t \right )} + \left (C_{2} + \frac {t}{4} + \frac {\log {\left (\sin {\left (4 t \right )} \right )}}{16}\right ) \sin {\left (4 t \right )}\right ) e^{- 3 t} \]