23.3.158 problem 160

Internal problem ID [5872]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 160
Date solved : Tuesday, September 30, 2025 at 02:05:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.128 (sec). Leaf size: 30

ode:=(a*cos(x)^2-sec(x)^2)*y(x)-tan(x)*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sec \left (x \right ) \left (c_1 \sinh \left (\sqrt {-a}\, \sin \left (x \right )\right )+c_2 \cosh \left (\sqrt {-a}\, \sin \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.862 (sec). Leaf size: 86

ode=(a*Cos[x]^2 - Sec[x]^2)*y[x] - Tan[x]*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sec (x) e^{\sqrt {a} \sqrt {\cos (x)-1} \sqrt {\cos (x)+1}} \left (c_2 \int _1^{\cos (x)}\frac {e^{-2 \sqrt {a} \sqrt {K[1]-1} \sqrt {K[1]+1}} K[1]}{\sqrt {1-K[1]^2}}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

False