23.3.165 problem 167

Internal problem ID [5879]
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 : 167
Date solved : Friday, October 03, 2025 at 01:44:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 60

ode:=b*y(x)+a*tan(x)*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \cos \left (x \right )^{\frac {a}{2}+\frac {1}{2}} \left (c_1 \operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {a}{2}+\frac {1}{2}, \sin \left (x \right )\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {a}{2}+\frac {1}{2}, \sin \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 129

ode=b*y[x] + a*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 c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-a-\sqrt {a^2+4 b}\right ),\frac {1}{4} \left (\sqrt {a^2+4 b}-a\right ),\frac {1-a}{2},\cos ^2(x)\right )+i^{a+1} c_2 \cos ^{a+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (a-\sqrt {a^2+4 b}+2\right ),\frac {1}{4} \left (a+\sqrt {a^2+4 b}+2\right ),\frac {a+3}{2},\cos ^2(x)\right ) \end{align*}

✗ Sympy

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

False