74.12.37 problem 37

Internal problem ID [16265]
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 : 37
Date solved : Thursday, March 13, 2025 at 08:08:26 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 42

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

\[ y = \sin \left (t \right ) \cos \left (t \right ) \ln \left (\cos \left (t \right )\right )+\left (2 c_{1} -t \right ) \cos \left (t \right )^{2}+\frac {\sin \left (t \right ) \left (4 c_{2} +1\right ) \cos \left (t \right )}{2}+\frac {t}{2}-c_{1} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 55

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

\[ y(t)\to \cos (2 t) \int _1^t-\sin ^2(K[1])dK[1]-\sin (t) \cos ^3(t)+c_1 \cos (2 t)+c_2 \sin (2 t)+\sin (t) \cos (t) \log (\cos (t)) \]

✓ Sympy. Time used: 0.792 (sec). Leaf size: 26

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

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