23.1.489 problem 479

Internal problem ID [5096]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 479
Date solved : Tuesday, September 30, 2025 at 11:36:11 AM
CAS classification : [_separable]

\begin{align*} 3 y y^{\prime }+5 \cot \left (x \right ) \cot \left (y\right ) \cos \left (y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 22

ode:=3*y(x)*diff(y(x),x)+5*cot(x)*cot(y(x))*cos(y(x))^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (\sin \left (x \right )\right )+c_1 -\frac {3 \tan \left (y\right )}{10}+\frac {3 y \sec \left (y\right )^{2}}{10} = 0 \]

✓ Mathematica. Time used: 0.416 (sec). Leaf size: 288

ode=3*y[x]*D[y[x],x]+5*Cot[x]*Cot[y[x]]*Cos[y[x]]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x5 \exp \left (\int _1^{y(x)}\frac {3}{5} \left (K[1] \sec ^2(K[1])+5\right ) \tan (K[1])dK[1]\right ) (\cos (K[2]-3 y(x))+3 \cos (K[2]-y(x))+3 \cos (K[2]+y(x))+\cos (K[2]+3 y(x)))dK[2]+\int _1^{y(x)}\left (12 \exp \left (\int _1^{K[3]}\frac {3}{5} \left (K[1] \sec ^2(K[1])+5\right ) \tan (K[1])dK[1]\right ) (\cos (x-K[3])-\cos (x+K[3])) K[3]-\int _1^x\left (5 \exp \left (\int _1^{K[3]}\frac {3}{5} \left (K[1] \sec ^2(K[1])+5\right ) \tan (K[1])dK[1]\right ) (3 \sin (K[2]-3 K[3])+3 \sin (K[2]-K[3])-3 \sin (K[2]+K[3])-3 \sin (K[2]+3 K[3]))+3 \exp \left (\int _1^{K[3]}\frac {3}{5} \left (K[1] \sec ^2(K[1])+5\right ) \tan (K[1])dK[1]\right ) (\cos (K[2]-3 K[3])+3 \cos (K[2]-K[3])+3 \cos (K[2]+K[3])+\cos (K[2]+3 K[3])) \left (K[3] \sec ^2(K[3])+5\right ) \tan (K[3])\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.450 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x)*Derivative(y(x), x) + 5*cos(y(x))**2/(tan(x)*tan(y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {y \tan {\left (y \right )}}{\cos ^{2}{\left (y \right )}}\, dy = C_{1} - \frac {5 \log {\left (\sin {\left (x \right )} \right )}}{3} \]