68.12.32 problem 32

Internal problem ID [17626]
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 : 32
Date solved : Thursday, October 02, 2025 at 02:26:25 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\tan \left (3 t \right )^{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+9*y(t) = tan(3*t)^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_1 -\frac {2}{9}+\frac {\sin \left (3 t \right ) \ln \left (\sec \left (3 t \right )+\tan \left (3 t \right )\right )}{9} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 71

ode=D[y[t],{t,2}]+9*y[t]==Tan[3*t]^2; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (3 t) \int _1^t-\frac {1}{3} \sin (3 K[1]) \tan ^2(3 K[1])dK[1]+\frac {1}{9} \sin (3 t) \text {arctanh}(\sin (3 t))-\frac {1}{9} \sin ^2(3 t)+c_1 \cos (3 t)+c_2 \sin (3 t) \end{align*}

✓ Sympy. Time used: 0.270 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) - tan(3*t)**2 + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \cos {\left (3 t \right )} + \left (C_{1} - \frac {\log {\left (\sin {\left (3 t \right )} - 1 \right )}}{18} + \frac {\log {\left (\sin {\left (3 t \right )} + 1 \right )}}{18}\right ) \sin {\left (3 t \right )} - \frac {2}{9} \]