47.4.28 problem 31

Internal problem ID [9802]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 06:42:01 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.073 (sec). Leaf size: 24

ode:=y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2*(1-diff(y(x),x)*sin(y(x))-y(x)*diff(y(x),x)*cos(y(x))); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \\ -\cos \left (y\right )+c_1 \ln \left (y\right )-x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 150

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {c_1-\int _1^{K[2]}(-\cos (K[1]) K[1]-\sin (K[1]))dK[1]}{K[2]}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {-c_1-\int _1^{K[2]}(-\cos (K[1]) K[1]-\sin (K[1]))dK[1]}{K[2]}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {c_1-\int _1^{K[2]}(-\cos (K[1]) K[1]-\sin (K[1]))dK[1]}{K[2]}dK[2]\&\right ][x+c_2] \end{align*}

✗ Sympy

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

Timed Out