83.3.11 problem 11

Internal problem ID [18988]
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, March 13, 2025 at 01:17:03 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: 0.081 (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);
 

\[ y \left (x \right ) = \sqrt {2}\, \sqrt {\frac {\sin \left (x \right ) x +c_{1}}{\operatorname {LambertW}\left (2 \sin \left (x \right ) x +2 c_{1} \right )}} \]

✓ Mathematica. Time used: 60.188 (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: 0.868 (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)
 

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