29.37.18 problem 1138

Internal problem ID [5676]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1138
Date solved : Monday, January 27, 2025 at 01:06:26 PM
CAS classification : [_Clairaut]

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

✓ Solution by Maple

Time used: 0.449 (sec). Leaf size: 139

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

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

✓ Solution by Mathematica

Time used: 0.342 (sec). Leaf size: 77

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