77.3.11 problem 11

Internal problem ID [20352]
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 : 11
Date solved : Thursday, October 02, 2025 at 05:44:56 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\sin \left (x \right )+x \cos \left (x \right )}{y \left (2 \ln \left (y\right )+1\right )} \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 29

ode:=diff(y(x),x) = (sin(x)+x*cos(x))/y(x)/(2*ln(y(x))+1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {2}\, \sqrt {\frac {x \sin \left (x \right )+c_1}{\operatorname {LambertW}\left (2 x \sin \left (x \right )+2 c_1 \right )}} \]

✓ Mathematica. Time used: 60.146 (sec). Leaf size: 72

ode=D[y[x],x]== (Sin[x]+x*Cos[x])/( y[x]*(2*Log[y[x]]+1)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {2} \sqrt {x \sin (x)+c_1}}{\sqrt {W(2 (x \sin (x)+c_1))}}\\ y(x)&\to \frac {\sqrt {2} \sqrt {x \sin (x)+c_1}}{\sqrt {W(2 (x \sin (x)+c_1))}} \end{align*}

✓ Sympy. Time used: 0.545 (sec). Leaf size: 15

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

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