83.49.14 problem Ex 14 page 131

Internal problem ID [19556]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VIII. Linear equations of second order
Problem number : Ex 14 page 131
Date solved : Monday, March 31, 2025 at 07:33:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+\frac {\csc \left (x \right )^{2} y}{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)+cot(x)*diff(y(x),x)+1/2*csc(x)^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (\csc \left (x \right )+\cot \left (x \right )\right )^{-\frac {i \sqrt {2}}{2}}+c_2 \left (\csc \left (x \right )+\cot \left (x \right )\right )^{\frac {i \sqrt {2}}{2}} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+Cot[x]*D[y[x],x]+1/2*Csc[x]^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\frac {\text {arctanh}(\cos (x))}{\sqrt {2}}\right )-c_2 \sin \left (\frac {\text {arctanh}(\cos (x))}{\sqrt {2}}\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE y(x)*tan(x)/(2*sin(x)**2) + tan(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method