##### 4.20.41 $$\left (\left (1-a^2\right ) x^2+y(x)^2\right ) y'(x)^2+2 a^2 x y(x) y'(x)+\left (1-a^2\right ) y(x)^2+x^2=0$$

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

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous ODE, $$x^n f\left ( \frac {y}{x} , y' \right )=0$$, Solve for $$p$$

Mathematica
cpu = 0.412667 (sec), leaf count = 93

$\left \{\text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \tan ^{-1}\left (\frac {y(x)}{x}\right )=\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )+\log (x)+c_1,y(x)\right ],\text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \tan ^{-1}\left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )+\log (x)=c_1,y(x)\right ]\right \}$

Maple
cpu = 3.215 (sec), leaf count = 65

$\left [y \left (x \right ) = \tan \left (\RootOf \left (-2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 \textit {\_C1} \right )\right ) x, y \left (x \right ) = \tan \left (\RootOf \left (2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\frac {x^{2}}{\cos \left (\textit {\_Z} \right )^{2}}\right )+2 \textit {\_C1} \right )\right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

{Solve[Sqrt[-1 + a]*Sqrt[1 + a]*ArcTan[y[x]/x] == C[1] + Log[x] + Log[1 + y[x]^2
/x^2]/2, y[x]], Solve[Sqrt[-1 + a]*Sqrt[1 + a]*ArcTan[y[x]/x] + Log[x] + Log[1 +
 y[x]^2/x^2]/2 == C[1], y[x]]}

Maple raw input

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

Maple raw output

[y(x) = tan(RootOf(-2*_Z*(a^2-1)^(1/2)-ln(x^2/cos(_Z)^2)+2*_C1))*x, y(x) = tan(R
ootOf(2*_Z*(a^2-1)^(1/2)-ln(x^2/cos(_Z)^2)+2*_C1))*x]