90.24.13 problem 13

Internal problem ID [25378]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 371
Problem number : 13
Date solved : Friday, October 03, 2025 at 08:08:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\cos \left (2 t \right )+1\right ) y^{\prime \prime }-4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (2 t \right )}{1+\cos \left (t \right )} \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 26

ode:=(cos(2*t)+1)*diff(diff(y(t),t),t)-4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \ln \left (\cos \left (t \right )+i \sin \left (t \right )\right ) c_2 \tan \left (t \right )+i c_2 +c_1 \tan \left (t \right ) \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 67

ode=(Cos[2*t]+1)*D[y[t],{t,2}]-4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {c_2 \sin ^2(t)^{3/4} \sec (t) \left (-\arctan \left (\frac {\cos (t)}{\sqrt {\sin ^2(t)}}\right )\right )+c_2 \sqrt [4]{\sin ^2(t)}+c_1 \sin ^2(t)^{3/4} \sec (t)}{\sqrt [4]{-\sin ^2(t)}} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((cos(2*t) + 1)*Derivative(y(t), (t, 2)) - 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False