83.27.13 problem 13

Internal problem ID [19288]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (A) at page 104
Problem number : 13
Date solved : Monday, March 31, 2025 at 07:05:45 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.066 (sec). Leaf size: 39

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

\[ y = -\ln \left (\cos \left (x \right )-1\right ) c_2 \sin \left (x \right )^{2}+\ln \left (\cos \left (x \right )+1\right ) c_2 \sin \left (x \right )^{2}+c_1 \sin \left (x \right )^{2}+2 \cos \left (x \right ) c_2 \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 45

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

\[ y(x)\to -c_1 \sin ^2(x)-\frac {1}{4} c_2 \left (2 \cos (x)+\sin ^2(x) (\log (\cos (x)+1)-\log (1-\cos (x)))\right ) \]

✗ Sympy

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

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