60.1.566 problem 569

Internal problem ID [10580]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 569
Date solved : Monday, January 27, 2025 at 09:16:31 PM
CAS classification : [_Clairaut]

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

✓ Solution by Maple

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

dsolve((diff(y(x),x)^2+1)*sin(x*diff(y(x),x)-y(x))^2-1=0,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*}

✓ Solution by Mathematica

Time used: 0.341 (sec). Leaf size: 77

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