68.12.11 problem 11

Internal problem ID [17605]
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 : 11
Date solved : Thursday, October 02, 2025 at 02:26:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+26 y&={\mathrm e}^{t} \left (\sec \left (5 t \right )+\csc \left (5 t \right )\right ) \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 52

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+26*y(t) = exp(t)*(sec(5*t)+csc(5*t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{t} \left (-\frac {\cos \left (5 t \right ) \ln \left (\cos \left (5 t \right )\right )}{5}-\frac {\sin \left (5 t \right ) \ln \left (\sin \left (5 t \right )\right )}{5}+\left (t -5 c_1 \right ) \cos \left (5 t \right )-\sin \left (5 t \right ) \left (t +5 c_2 \right )\right )}{5} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 50

ode=D[y[t],{t,2}]-2*D[y[t],t]+26*y[t]==Exp[t]*(Sec[5*t]+Csc[5*t]); 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{25} e^t (\cos (5 t) (-5 t+\log (\cos (5 t))+25 c_2)+\sin (5 t) (\log (\sin (5 t))+5 (t+5 c_1))) \end{align*}

✓ Sympy. Time used: 0.487 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((-1/cos(5*t) - 1/sin(5*t))*exp(t) + 26*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - \frac {t}{5} + \frac {\log {\left (\cos {\left (5 t \right )} \right )}}{25}\right ) \cos {\left (5 t \right )} + \left (C_{2} + \frac {t}{5} + \frac {\log {\left (\sin {\left (5 t \right )} \right )}}{25}\right ) \sin {\left (5 t \right )}\right ) e^{t} \]