23.2.284 problem 303

Internal problem ID [5639]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 303
Date solved : Tuesday, September 30, 2025 at 01:24:00 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}+\left (\cos \left (x \right ) \cot \left (x \right )-y\right ) {y^{\prime }}^{2}-\left (1+y \cos \left (x \right ) \cot \left (x \right )\right ) y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 32

ode:=diff(y(x),x)^3+(cos(x)*cot(x)-y(x))*diff(y(x),x)^2-(1+y(x)*cos(x)*cot(x))*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \,{\mathrm e}^{x} \\ y &= -\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_1 \\ y &= -\cos \left (x \right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 38

ode=(D[y[x],x])^3 +(Cos[x]*Cot[x]-y[x])*(D[y[x],x])^2-(1+y[x]*Cos[x]*Cot[x])*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^x\\ y(x)&\to \text {arctanh}(\cos (x))+c_1\\ y(x)&\to \int _1^x\sin (K[1])dK[1]+c_1 \end{align*}

✓ Sympy. Time used: 4.240 (sec). Leaf size: 102

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x)*cos(x)/tan(x) - 1)*Derivative(y(x), x) + (-y(x) + cos(x)/tan(x))*Derivative(y(x), x)**2 + y(x) + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} e^{x}, \ y{\left (x \right )} = C_{1} - \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4} - \frac {\cos {\left (x \right )}}{2} + \frac {\int \frac {\sqrt {\cos ^{2}{\left (x \right )} + 4 \tan ^{2}{\left (x \right )}}}{\tan {\left (x \right )}}\, dx}{2}, \ y{\left (x \right )} = C_{1} - \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} + \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4} - \frac {\cos {\left (x \right )}}{2} - \frac {\int \frac {\sqrt {\cos ^{2}{\left (x \right )} + 4 \tan ^{2}{\left (x \right )}}}{\tan {\left (x \right )}}\, dx}{2}\right ] \]