68.12.12 problem 12

Internal problem ID [17606]
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 : 12
Date solved : Thursday, October 02, 2025 at 02:26:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+12 y^{\prime }+37 y&={\mathrm e}^{-6 t} \csc \left (t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+12*diff(y(t),t)+37*y(t) = exp(-6*t)*csc(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\left (\ln \left (\csc \left (t \right )\right ) \sin \left (t \right )+\left (t -c_1 \right ) \cos \left (t \right )-c_2 \sin \left (t \right )\right ) {\mathrm e}^{-6 t} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 30

ode=D[y[t],{t,2}]+12*D[y[t],t]+37*y[t]==Exp[-6*t]*Csc[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-6 t} ((-t+c_2) \cos (t)+\sin (t) (\log (\sin (t))+c_1)) \end{align*}

✓ Sympy. Time used: 0.304 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(37*y(t) + 12*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-6*t)/sin(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\left (C_{1} - t\right ) \cos {\left (t \right )} + \left (C_{2} + \log {\left (\sin {\left (t \right )} \right )}\right ) \sin {\left (t \right )}\right ) e^{- 6 t} \]