60.1.556 problem 569

Internal problem ID [10570]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 569
Date solved : Wednesday, March 05, 2025 at 12:08:03 PM
CAS classification : [_Clairaut]

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

✓ Maple. Time used: 0.429 (sec). Leaf size: 139

ode:=(1+diff(y(x),x)^2)*sin(-y(x)+x*diff(y(x),x))^2-1 = 0; 
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: 0.341 (sec). Leaf size: 77

ode=-1 + Sin[y[x] - x*D[y[x],x]]^2*(1 + D[y[x],x]^2)==0; 
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