##### 4.20.38 $$\left (a^2 x^2-y(x)^2\right ) y'(x)^2+\left (a^2-1\right ) x^2-2 x y(x) y'(x)=0$$

ODE
$\left (a^2 x^2-y(x)^2\right ) y'(x)^2+\left (a^2-1\right ) x^2-2 x y(x) y'(x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y$$

Mathematica
cpu = 0.279551 (sec), leaf count = 49

$\left \{\left \{y(x)\to a c_1-\sqrt {-x^2+c_1{}^2}\right \},\left \{y(x)\to a c_1+\sqrt {-x^2+c_1{}^2}\right \}\right \}$

Maple
cpu = 0.353 (sec), leaf count = 173

$\left [y \left (x \right ) = \sqrt {a^{2}-1}\, x, y \left (x \right ) = -\sqrt {a^{2}-1}\, x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{3}+\textit {\_a} \,a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-a^{4}+a^{2}}-\textit {\_a}}{\textit {\_a}^{4}-\textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}-a^{2}+1}d \textit {\_a} +\textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-\textit {\_a} \,a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-a^{4}+a^{2}}+\textit {\_a}}{\textit {\_a}^{4}-\textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}-a^{2}+1}d \textit {\_a} \right )+\textit {\_C1} \right ) x\right ]$ Mathematica raw input

DSolve[(-1 + a^2)*x^2 - 2*x*y[x]*y'[x] + (a^2*x^2 - y[x]^2)*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> a*C[1] - Sqrt[-x^2 + C[1]^2]}, {y[x] -> a*C[1] + Sqrt[-x^2 + C[1]^2]}}

Maple raw input

dsolve((a^2*x^2-y(x)^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+(a^2-1)*x^2 = 0, y(x))

Maple raw output

[y(x) = (a^2-1)^(1/2)*x, y(x) = -(a^2-1)^(1/2)*x, y(x) = RootOf(-ln(x)+Intat((-_
a^3+_a*a^2+(_a^2*a^2-a^4+a^2)^(1/2)-_a)/(_a^4-_a^2*a^2+2*_a^2-a^2+1),_a = _Z)+_C
1)*x, y(x) = RootOf(-ln(x)-Intat((_a^3-_a*a^2+(_a^2*a^2-a^4+a^2)^(1/2)+_a)/(_a^4
-_a^2*a^2+2*_a^2-a^2+1),_a = _Z)+_C1)*x]