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 = 1.54616 (sec), leaf count = 71
\[\left \{\left \{y(x)\to \frac {c c_1-\sqrt {b^2-c^2 \left (x-c_1\right ){}^2}}{a}\right \},\left \{y(x)\to \frac {\sqrt {b^2-c^2 \left (x-c_1\right ){}^2}+c c_1}{a}\right \}\right \}\]
Maple ✓
cpu = 1.334 (sec), leaf count = 195
\[ \left \{ {\frac {{a}^{2} \left ( y \left ( x \right ) \right ) ^{2}-2\,y \left ( x \right ) acx+{c}^{2}{x}^{2}-2\,{b}^{2}}{{a}^{2}}}=0,x-\int ^{y \left ( x \right ) -{\frac {cx}{a}}}\!-{\frac {a}{ \left ( 2\,{{\it \_a}}^{2}{a}^{2}-4\,{b}^{2} \right ) c} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{b}^{2}-\sqrt {-{a}^{2}{{\it \_a}}^{2} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{b}^{2} \right ) } \right ) }{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) -{\frac {cx}{a}}}\!-{\frac {a}{ \left ( 2\,{{\it \_a}}^{2}{a}^{2}-4\,{b}^{2} \right ) c} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{b}^{2}+\sqrt {-{a}^{2}{{\it \_a}}^{2} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{b}^{2} \right ) } \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] 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),'implicit')
Maple raw output
(a^2*y(x)^2-2*y(x)*a*c*x+c^2*x^2-2*b^2)/a^2 = 0, 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 = y(x)-1/a*c*x)-_C1 = 0, 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 = y(x)-1/a*c*x)-_C1 = 0