ODE
\[ 2 a^2 y'(x)+\left (a^2-(x-y(x))^2\right ) y'(x)^2+a^2-(x-y(x))^2=0 \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 94.1786 (sec), leaf count = 53
\[\left \{\left \{y(x)\to c_1-\sqrt {a^2-\left (x-c_1\right ){}^2}\right \},\left \{y(x)\to \sqrt {a^2-\left (x-c_1\right ){}^2}+c_1\right \}\right \}\]
Maple ✓
cpu = 0.415 (sec), leaf count = 137
\[ \left \{ x-\int ^{y \left ( x \right ) -x}\!{\frac {1}{-2\,{{\it \_a}}^{2}+4\,{a}^{2}} \left ( -2\,{a}^{2}+{{\it \_a}}^{2}-\sqrt {-{{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{a}^{2}} \right ) }{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) -x}\!{\frac {1}{-2\,{{\it \_a}}^{2}+4\,{a}^{2}} \left ( -2\,{a}^{2}+{{\it \_a}}^{2}+\sqrt {-{{\it \_a}}^{4}+2\,{{\it \_a}}^{2}{a}^{2}} \right ) }{d{\it \_a}}-{\it \_C1}=0, \left ( y \left ( x \right ) \right ) ^{2}-2\,xy \left ( x \right ) -2\,{a}^{2}+{x}^{2}=0 \right \} \] Mathematica raw input
DSolve[a^2 - (x - y[x])^2 + 2*a^2*y'[x] + (a^2 - (x - y[x])^2)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> -Sqrt[a^2 - (x - C[1])^2] + C[1]}, {y[x] -> Sqrt[a^2 - (x - C[1])^2] + C[1]}}
Maple raw input
dsolve((a^2-(x-y(x))^2)*diff(y(x),x)^2+2*a^2*diff(y(x),x)+a^2-(x-y(x))^2 = 0, y(x),'implicit')
Maple raw output
y(x)^2-2*x*y(x)-2*a^2+x^2 = 0, x-Intat((-2*a^2+_a^2+(-_a^4+2*_a^2*a^2)^(1/2))/(-2*_a^2+4*a^2),_a = y(x)-x)-_C1 = 0, x-Intat((-2*a^2+_a^2-(-_a^4+2*_a^2*a^2)^(1/2))/(-2*_a^2+4*a^2),_a = y(x)-x)-_C1 = 0