83.3.10 problem 10

Internal problem ID [18987]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (B) at page 9
Problem number : 10
Date solved : Thursday, March 13, 2025 at 01:16:56 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.573 (sec). Leaf size: 49

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

\[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (\left (x^{2} \cos \left (x \right )-2 \sin \left (x \right ) x -2 \cos \left (x \right )-c_{1} \right ) {\mathrm e}^{-1}\right )}{x^{2} \cos \left (x \right )-2 \sin \left (x \right ) x -2 \cos \left (x \right )-c_{1}} \]

✓ Mathematica. Time used: 60.095 (sec). Leaf size: 45

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

\[ y(x)\to \frac {W\left (\frac {\left (x^2-2\right ) \cos (x)-2 x \sin (x)+c_1}{e}\right )}{\left (x^2-2\right ) \cos (x)-2 x \sin (x)+c_1} \]

✓ Sympy. Time used: 20.430 (sec). Leaf size: 119

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

\[ y{\left (x \right )} = \frac {2 e^{i x} W\left (\frac {\left (2 C_{1} e^{i x} + x^{2} e^{2 i x} + x^{2} + 2 i x e^{2 i x} - 2 i x - 2 e^{2 i x} - 2\right ) e^{- i x - 1}}{2}\right )}{2 C_{1} e^{i x} + x^{2} e^{2 i x} + x^{2} + 2 i x e^{2 i x} - 2 i x - 2 e^{2 i x} - 2} \]