83.38.6 problem 6

Internal problem ID [19394]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 6
Date solved : Monday, March 31, 2025 at 07:12:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

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

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

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 21

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

\[ y(x)\to c_1 \cos (\text {arctanh}(\cos (x)))-c_2 \sin (\text {arctanh}(\cos (x))) \]

✗ Sympy

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

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