68.12.35 problem 35

Internal problem ID [17629]
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 : 35
Date solved : Thursday, October 02, 2025 at 02:26:28 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\tan \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)+4*y(t) = tan(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 -\frac {\cos \left (2 t \right ) \ln \left (\sec \left (2 t \right )+\tan \left (2 t \right )\right )}{4} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 40

ode=D[y[t],{t,2}]+4*y[t]==Tan[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {1}{4} \cos (2 t) \text {arctanh}(\sin (2 t))+c_1 \cos (2 t)+\frac {1}{4} (-1+4 c_2) \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.266 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - tan(2*t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} \sin {\left (2 t \right )} + \left (C_{1} + \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{8} - \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{8}\right ) \cos {\left (2 t \right )} \]