68.12.9 problem 9

Internal problem ID [17603]
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 : 9
Date solved : Thursday, October 02, 2025 at 02:26:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+50 y&={\mathrm e}^{-t} \csc \left (7 t \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+50*y(t) = exp(-t)*csc(7*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {\left (\frac {\ln \left (\csc \left (7 t \right )\right ) \sin \left (7 t \right )}{7}+\left (t -7 c_1 \right ) \cos \left (7 t \right )-7 c_2 \sin \left (7 t \right )\right ) {\mathrm e}^{-t}}{7} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 42

ode=D[y[t],{t,2}]+2*D[y[t],t]+50*y[t]==Exp[-t]*Csc[7*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{49} e^{-t} (\sin (7 t) (\log (\sin (7 t))+49 c_1)-7 (t-7 c_2) \cos (7 t)) \end{align*}

✓ Sympy. Time used: 0.330 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(50*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-t)/sin(7*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - \frac {t}{7}\right ) \cos {\left (7 t \right )} + \left (C_{2} + \frac {\log {\left (\sin {\left (7 t \right )} \right )}}{49}\right ) \sin {\left (7 t \right )}\right ) e^{- t} \]