23.2.347 problem 396

Internal problem ID [5702]
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 : 396
Date solved : Tuesday, September 30, 2025 at 02:00:44 PM
CAS classification : [_Clairaut]

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

✓ Maple. Time used: 0.103 (sec). Leaf size: 139

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

\begin{align*} y &= -x \sqrt {\frac {1}{x}}\, \sqrt {1-x}-\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \\ y &= x \sqrt {\frac {1}{x}}\, \sqrt {1-x}+\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \\ y &= -x \sqrt {-\frac {1}{x}}\, \sqrt {x +1}+\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \\ y &= x \sqrt {-\frac {1}{x}}\, \sqrt {x +1}-\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \\ y &= c_1 x -\arcsin \left (\frac {1}{\sqrt {c_1^{2}+1}}\right ) \\ y &= c_1 x +\arcsin \left (\frac {1}{\sqrt {c_1^{2}+1}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.204 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to c_1 x-\frac {1}{2} \arccos \left (\frac {-1+c_1{}^2}{1+c_1{}^2}\right )\\ y(x)&\to \frac {1}{2} \arccos \left (\frac {-1+c_1{}^2}{1+c_1{}^2}\right )+c_1 x\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [_X0, sin(_X0*x - y(x))] 
No algorithms are implemented to solve equation _X0**2*sin(_X0*x - y(x))**2 + sin(_X0*x - y(x))**2 - 1