4.21.3 $a^2{y}^{\prime }(x{)}^2(b^2-(cx-ay(x){)}^2)+c^2(b^2-(cx-ay(x){)}^2)+2a{b}^{2}c{y}^{\prime }(x)=0$

ODE
$$a^2{y}^{\prime }(x{)}^2(b^2-(cx-ay(x){)}^2)+c^2(b^2-(cx-ay(x){)}^2)+2a{b}^{2}c{y}^{\prime }(x)=0$$ 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

$$\{\{y(x)\to \frac{c{c}_1-\sqrt{b^2-c^2(x-c_1){}^2}}{a}\},\{y(x)\to \frac{\sqrt{b^2-c^2(x-c_1){}^2}+c{c}_1}{a}\}\}$$

Maple ✓
cpu = 1.334 (sec), leaf count = 195

$$\{\frac{a^2{\left(y\left(x\right)\right)}^2-2y\left(x\right)acx+c^2{x}^2-2b^2}{a^2}=0,x-{\int }^{y\left(x\right)-\frac{cx}{a}}-\frac{a}{(2{\mathit{\_}\mathit{a}}^2{a}^2-4b^2)c}({\mathit{\_}\mathit{a}}^2{a}^2-2b^2-\sqrt{-a^2{\mathit{\_}\mathit{a}}^2({\mathit{\_}\mathit{a}}^2{a}^2-2b^2)})d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)-\frac{cx}{a}}-\frac{a}{(2{\mathit{\_}\mathit{a}}^2{a}^2-4b^2)c}({\mathit{\_}\mathit{a}}^2{a}^2-2b^2+\sqrt{-a^2{\mathit{\_}\mathit{a}}^2({\mathit{\_}\mathit{a}}^2{a}^2-2b^2)})d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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