68.12.14 problem 14

Internal problem ID [17608]
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 : 14
Date solved : Thursday, October 02, 2025 at 02:26:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-10 y^{\prime }+34 y&={\mathrm e}^{5 t} \cot \left (3 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 40

ode:=diff(diff(y(t),t),t)-10*diff(y(t),t)+34*y(t) = exp(5*t)*cot(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{5 t} \left (c_2 \sin \left (3 t \right )+c_1 \cos \left (3 t \right )+\frac {\sin \left (3 t \right ) \ln \left (\csc \left (3 t \right )-\cot \left (3 t \right )\right )}{9}\right ) \]

✓ Mathematica. Time used: 0.093 (sec). Leaf size: 73

ode=D[y[t],{t,2}]-10*D[y[t],t]+34*y[t]==Exp[5*t]*Cot[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{5 t} \left (\cos (3 t) \int _1^t-\frac {1}{3} \cos (3 K[2])dK[2]+\sin (3 t) \int _1^t\frac {1}{3} \cos (3 K[1]) \cot (3 K[1])dK[1]+c_2 \cos (3 t)+c_1 \sin (3 t)\right ) \end{align*}

✓ Sympy. Time used: 0.381 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(34*y(t) - exp(5*t)/tan(3*t) - 10*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{2} \cos {\left (3 t \right )} + \left (C_{1} + \frac {\log {\left (\cos {\left (3 t \right )} - 1 \right )}}{18} - \frac {\log {\left (\cos {\left (3 t \right )} + 1 \right )}}{18}\right ) \sin {\left (3 t \right )}\right ) e^{5 t} \]