23.3.587 problem 595

Internal problem ID [6301]
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 : 595
Date solved : Friday, October 03, 2025 at 02:00:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (\operatorname {a0} +\operatorname {a1} \cos \left (x \right )^{2}\right ) y+a^{2} \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1-a^{2} \cos \left (x \right )^{2}\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.284 (sec). Leaf size: 212

ode:=(a0+a1*cos(x)^2)*y(x)+a^2*cos(x)*sin(x)*diff(y(x),x)+(1-a^2*cos(x)^2)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sqrt {a^{2} \cos \left (2 x \right )+a^{2}-2}\, \operatorname {HeunG}\left (\frac {1}{a^{2}}, \frac {a^{2}-\operatorname {a0}}{4 a^{2}}, \frac {3 a +\sqrt {a^{2}-4 \operatorname {a1}}}{4 a}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, a +a^{2}+2 \operatorname {a1}}{2 a \left (a +\sqrt {a^{2}-4 \operatorname {a1}}\right )}, \frac {1}{2}, \frac {1}{2}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )+\frac {c_2 \cos \left (x \right )^{{3}/{2}} \sqrt {\sin \left (x \right ) \left (a^{2} \cos \left (2 x \right )+a^{2}-2\right )}\, \operatorname {HeunG}\left (\frac {1}{a^{2}}, \frac {4 a^{2}-\operatorname {a0} +1}{4 a^{2}}, \frac {5 a +\sqrt {a^{2}-4 \operatorname {a1}}}{4 a}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, a +a^{2}+\operatorname {a1}}{a \left (a +\sqrt {a^{2}-4 \operatorname {a1}}\right )}, \frac {3}{2}, \frac {1}{2}, \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )}{\sqrt {\sin \left (2 x \right )}} \]

✗ Mathematica

ode=(a0 + a1*Cos[x]^2)*y[x] + a^2*Cos[x]*Sin[x]*D[y[x],x] + (1 - a^2*Cos[x]^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
a0 = symbols("a0") 
a1 = symbols("a1") 
y = Function("y") 
ode = Eq(a**2*sin(x)*cos(x)*Derivative(y(x), x) + (a0 + a1*cos(x)**2)*y(x) + (-a**2*cos(x)**2 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False