90.24.11 problem 11

Internal problem ID [25376]
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 : 11
Date solved : Friday, October 03, 2025 at 12:00:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\tan \left (t \right ) y^{\prime }-\sec \left (t \right )^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-tan(t)*diff(y(t),t)-sec(t)^2*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\sec \left (t \right ) \left (-c_1 -c_2 +\sin \left (t \right ) \left (-c_2 +c_1 \right )\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 27

ode=D[y[t],{t,2}]-Tan[t]*D[y[t],t]-Sec[t]^2*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {i c_2 \csc (t)+c_1}{\sqrt {-\cot ^2(t)}} \end{align*}

✗ Sympy

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

False