89.33.28 problem 31

Internal problem ID [25012]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 31
Date solved : Thursday, October 02, 2025 at 11:46:55 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 )-\cos \left (y\right ) y y^{\prime }\right ) \end{align*}

✓ Maple. Time used: 0.052 (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: 0.301 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to \text {InverseFunction}[-\cos (\text {$\#$1})+c_1 \log (\text {$\#$1})\&][x+c_2]\\ y(x)&\to \text {InverseFunction}[-\cos (\text {$\#$1})-c_1 \log (\text {$\#$1})\&][x+c_2]\\ y(x)&\to \text {InverseFunction}[-\cos (\text {$\#$1})+c_1 \log (\text {$\#$1})\&][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