ODE
\[ a^2 y'(x)^2 \left (b^2-(c x-a y(x))^2\right )+c^2 \left (b^2-(c x-a y(x))^2\right )+2 a b^2 c y'(x)=0 \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 1.37748 (sec), leaf count = 71
\[\left \{\left \{y(x)\to \frac {c c_1-\sqrt {b^2-c^2 (x-c_1){}^2}}{a}\right \},\left \{y(x)\to \frac {\sqrt {b^2-c^2 (x-c_1){}^2}+c c_1}{a}\right \}\right \}\]
Maple ✓
cpu = 0.759 (sec), leaf count = 195
\[\left [y \left (x \right ) = \frac {c x -\sqrt {2}\, b}{a}, y \left (x \right ) = \frac {c x +\sqrt {2}\, b}{a}, y \left (x \right ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {a \left (\textit {\_a}^{2} a^{2}-2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right ) c}d \textit {\_a} +\textit {\_C1} \right ) a +c x}{a}, y \left (x \right ) = \frac {\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {a \left (-\textit {\_a}^{2} a^{2}+2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}\right )}{2 \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right ) c}d \textit {\_a} +\textit {\_C1} \right ) a +c x}{a}\right ]\] Mathematica raw input
DSolve[c^2*(b^2 - (c*x - a*y[x])^2) + 2*a*b^2*c*y'[x] + a^2*(b^2 - (c*x - a*y[x])^2)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-Sqrt[b^2 - c^2*(x - C[1])^2] + c*C[1])/a}, {y[x] -> (Sqrt[b^2 - c^2*(x - C[1])^2] + c*C[1])/a}}
Maple raw input
dsolve(a^2*(b^2-(c*x-a*y(x))^2)*diff(y(x),x)^2+2*a*b^2*c*diff(y(x),x)+c^2*(b^2-(c*x-a*y(x))^2) = 0, y(x))
Maple raw output
[y(x) = (c*x-2^(1/2)*b)/a, y(x) = (c*x+2^(1/2)*b)/a, y(x) = (RootOf(-x+Intat(-1/2*a*(_a^2*a^2-2*b^2+(-a^2*_a^2*(_a^2*a^2-2*b^2))^(1/2))/(_a^2*a^2-2*b^2)/c,_a = _Z)+_C1)*a+c*x)/a, y(x) = (RootOf(-x+Intat(1/2*a*(-_a^2*a^2+2*b^2+(-a^2*_a^2*(_a^2*a^2-2*b^2))^(1/2))/(_a^2*a^2-2*b^2)/c,_a = _Z)+_C1)*a+c*x)/a]