75.12.25 problem 299

Internal problem ID [16812]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 299
Date solved : Thursday, March 13, 2025 at 08:51:48 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`], [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.653 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.296 (sec). Leaf size: 96

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

\[ \text {Solve}\left [\int _1^{\frac {\left (-\cos ^3(x)\right )^{2/3} \sec ^2(x) (\sin (x)+4 y(x))}{\sqrt [3]{2} (\sin (x)-2 y(x))}}\frac {2}{2 K[1]^3+3 \sqrt [3]{-2} K[1]+2}dK[1]=\frac {1}{9} 2^{2/3} \left (-\cos ^3(x)\right )^{2/3} \sec ^2(x) \log (\sin (x))+c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.987 (sec). Leaf size: 32

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

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