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 = 24.378 (sec), leaf count = 1158
\[\left \{\text {Solve}\left [c_1=\int _1^x\frac {e^{\frac {\tan ^{-1}\left (\frac {K[1]-y(x)}{\sqrt {(K[1]-y(x))^2-2 a^2}}\right ) (K[1]-y(x)) \sqrt {-2 a^2+K[1]^2+y(x)^2-2 K[1] y(x)}}{\sqrt {-(K[1]-y(x))^2 \left (-2 a^2+K[1]^2+y(x)^2-2 K[1] y(x)\right )}}} \sqrt {-a^2+K[1]^2+y(x)^2-2 K[1] y(x)} \left (a^2-\sqrt {-(K[1]-y(x))^2 \left (-2 a^2+K[1]^2+y(x)^2-2 K[1] y(x)\right )}\right )}{(a+K[1]-y(x)) (a-K[1]+y(x)) \sqrt {-2 a^2+K[1]^2+y(x)^2-2 K[1] y(x)}}dK[1]+\int _1^{y(x)}\left (\frac {e^{\frac {\tan ^{-1}\left (\frac {x-K[2]}{\sqrt {(x-K[2])^2-2 a^2}}\right ) \sqrt {(x-K[2])^2-2 a^2} (x-K[2])}{\sqrt {2 a^2 (x-K[2])^2-(x-K[2])^4}}} \sqrt {(x-K[2])^2-a^2}}{\sqrt {(x-K[2])^2-2 a^2}}-\int _1^x\frac {a^2 e^{\frac {\tan ^{-1}\left (\frac {K[1]-K[2]}{\sqrt {(K[1]-K[2])^2-2 a^2}}\right ) \sqrt {(K[1]-K[2])^2-2 a^2} (K[1]-K[2])}{\sqrt {2 a^2 (K[1]-K[2])^2-(K[1]-K[2])^4}}} (K[1]-K[2]) \left (2 a^2-K[1]^2-K[2]^2+2 K[1] K[2]-\sqrt {2 a^2 (K[1]-K[2])^2-(K[1]-K[2])^4}\right )}{\left ((K[1]-K[2])^2-2 a^2\right )^{3/2} \sqrt {(K[1]-K[2])^2-a^2} \sqrt {2 a^2 (K[1]-K[2])^2-(K[1]-K[2])^4}}dK[1]\right )dK[2],y(x)\right ],\text {Solve}\left [c_1=\int _1^x\frac {e^{\frac {\tan ^{-1}\left (\frac {K[3]-y(x)}{\sqrt {(K[3]-y(x))^2-2 a^2}}\right ) (y(x)-K[3]) \sqrt {-2 a^2+K[3]^2+y(x)^2-2 K[3] y(x)}}{\sqrt {-(K[3]-y(x))^2 \left (-2 a^2+K[3]^2+y(x)^2-2 K[3] y(x)\right )}}} \sqrt {-a^2+K[3]^2+y(x)^2-2 K[3] y(x)} \left (a^2+\sqrt {-(K[3]-y(x))^2 \left (-2 a^2+K[3]^2+y(x)^2-2 K[3] y(x)\right )}\right )}{(a+K[3]-y(x)) (a-K[3]+y(x)) \sqrt {-2 a^2+K[3]^2+y(x)^2-2 K[3] y(x)}}dK[3]+\int _1^{y(x)}\left (\frac {e^{-\frac {\tan ^{-1}\left (\frac {x-K[4]}{\sqrt {(x-K[4])^2-2 a^2}}\right ) \sqrt {(x-K[4])^2-2 a^2} (x-K[4])}{\sqrt {2 a^2 (x-K[4])^2-(x-K[4])^4}}} \sqrt {(x-K[4])^2-a^2}}{\sqrt {(x-K[4])^2-2 a^2}}-\int _1^x-\frac {a^2 e^{-\frac {\tan ^{-1}\left (\frac {K[3]-K[4]}{\sqrt {(K[3]-K[4])^2-2 a^2}}\right ) \sqrt {(K[3]-K[4])^2-2 a^2} (K[3]-K[4])}{\sqrt {2 a^2 (K[3]-K[4])^2-(K[3]-K[4])^4}}} (K[3]-K[4]) \left (2 a^2-K[3]^2-K[4]^2+2 K[3] K[4]+\sqrt {2 a^2 (K[3]-K[4])^2-(K[3]-K[4])^4}\right )}{\left ((K[3]-K[4])^2-2 a^2\right )^{3/2} \sqrt {(K[3]-K[4])^2-a^2} \sqrt {2 a^2 (K[3]-K[4])^2-(K[3]-K[4])^4}}dK[3]\right )dK[4],y(x)\right ]\right \}\]
Maple ✓
cpu = 0.508 (sec), leaf count = 130
\[\left [y \left (x \right ) = x -\sqrt {2}\, a, y \left (x \right ) = x +\sqrt {2}\, a, y \left (x \right ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{2} \left (\textit {\_a}^{2}-2 a^{2}\right )}}{2 \left (\textit {\_a}^{2}-2 a^{2}\right )}d \textit {\_a} +\textit {\_C1} \right ), y \left (x \right ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {2 a^{2}-\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2} \left (\textit {\_a}^{2}-2 a^{2}\right )}}{2 \textit {\_a}^{2}-4 a^{2}}d \textit {\_a} +\textit {\_C1} \right )\right ]\] 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
{Solve[C[1] == Inactive[Integrate][(E^((ArcTan[(K[1] - y[x])/Sqrt[-2*a^2 + (K[1] - y[x])^2]]*(K[1] - y[x])*Sqrt[-2*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2])/Sqrt[-((K[1] - y[x])^2*(-2*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2))])*Sqrt[-a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2]*(a^2 - Sqrt[-((K[1] - y[x])^2*(-2*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2))]))/((a + K[1] - y[x])*(a - K[1] + y[x])*Sqrt[-2*a^2 + K[1]^2 - 2*K[1]*y[x] + y[x]^2]), {K[1], 1, x}] + Inactive[Integrate][(E^((ArcTan[(x - K[2])/Sqrt[-2*a^2 + (x - K[2])^2]]*Sqrt[-2*a^2 + (x - K[2])^2]*(x - K[2]))/Sqrt[2*a^2*(x - K[2])^2 - (x - K[2])^4])*Sqrt[-a^2 + (x - K[2])^2])/Sqrt[-2*a^2 + (x - K[2])^2] - Inactive[Integrate][(a^2*E^((ArcTan[(K[1] - K[2])/Sqrt[-2*a^2 + (K[1] - K[2])^2]]*Sqrt[-2*a^2 + (K[1] - K[2])^2]*(K[1] - K[2]))/Sqrt[2*a^2*(K[1] - K[2])^2 - (K[1] - K[2])^4])*(K[1] - K[2])*(2*a^2 - K[1]^2 - Sqrt[2*a^2*(K[1] - K[2])^2 - (K[1] - K[2])^4] + 2*K[1]*K[2] - K[2]^2))/((-2*a^2 + (K[1] - K[2])^2)^(3/2)*Sqrt[-a^2 + (K[1] - K[2])^2]*Sqrt[2*a^2*(K[1] - K[2])^2 - (K[1] - K[2])^4]), {K[1], 1, x}], {K[2], 1, y[x]}], y[x]], Solve[C[1] == Inactive[Integrate][(E^((ArcTan[(K[3] - y[x])/Sqrt[-2*a^2 + (K[3] - y[x])^2]]*(-K[3] + y[x])*Sqrt[-2*a^2 + K[3]^2 - 2*K[3]*y[x] + y[x]^2])/Sqrt[-((K[3] - y[x])^2*(-2*a^2 + K[3]^2 - 2*K[3]*y[x] + y[x]^2))])*Sqrt[-a^2 + K[3]^2 - 2*K[3]*y[x] + y[x]^2]*(a^2 + Sqrt[-((K[3] - y[x])^2*(-2*a^2 + K[3]^2 - 2*K[3]*y[x] + y[x]^2))]))/((a + K[3] - y[x])*(a - K[3] + y[x])*Sqrt[-2*a^2 + K[3]^2 - 2*K[3]*y[x] + y[x]^2]), {K[3], 1, x}] + Inactive[Integrate][Sqrt[-a^2 + (x - K[4])^2]/(E^((ArcTan[(x - K[4])/Sqrt[-2*a^2 + (x - K[4])^2]]*Sqrt[-2*a^2 + (x - K[4])^2]*(x - K[4]))/Sqrt[2*a^2*(x - K[4])^2 - (x - K[4])^4])*Sqrt[-2*a^2 + (x - K[4])^2]) - Inactive[Integrate][-((a^2*(K[3] - K[4])*(2*a^2 - K[3]^2 + Sqrt[2*a^2*(K[3] - K[4])^2 - (K[3] - K[4])^4] + 2*K[3]*K[4] - K[4]^2))/(E^((ArcTan[(K[3] - K[4])/Sqrt[-2*a^2 + (K[3] - K[4])^2]]*Sqrt[-2*a^2 + (K[3] - K[4])^2]*(K[3] - K[4]))/Sqrt[2*a^2*(K[3] - K[4])^2 - (K[3] - K[4])^4])*(-2*a^2 + (K[3] - K[4])^2)^(3/2)*Sqrt[-a^2 + (K[3] - K[4])^2]*Sqrt[2*a^2*(K[3] - K[4])^2 - (K[3] - K[4])^4])), {K[3], 1, x}], {K[4], 1, y[x]}], y[x]]}
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))
Maple raw output
[y(x) = x-2^(1/2)*a, y(x) = x+2^(1/2)*a, y(x) = x+RootOf(-x+Intat(-1/2*(_a^2-2*a^2+(-_a^2*(_a^2-2*a^2))^(1/2))/(_a^2-2*a^2),_a = _Z)+_C1), y(x) = x+RootOf(-x+Intat(1/2*(2*a^2-_a^2+(-_a^2*(_a^2-2*a^2))^(1/2))/(_a^2-2*a^2),_a = _Z)+_C1)]