4.20.41 $((1-a^2)x^2+y(x{)}^2)y^{\prime }(x{)}^2+2a^{2}xy(x)y^{\prime }(x)+(1-a^2)y(x{)}^2+x^2=0$

ODE
$$((1-a^2)x^2+y(x{)}^2)y^{\prime }(x{)}^2+2a^{2}xy(x)y^{\prime }(x)+(1-a^2)y(x{)}^2+x^2=0$$ ODE Classification

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

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

Mathematica ✓
cpu = 0.176591 (sec), leaf count = 93

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

Maple ✓
cpu = 0.545 (sec), leaf count = 78

$$\{\ln \left(x\right)+\frac{1}{2}\ln \left(\frac{x^2+{\left(y\left(x\right)\right)}^2}{x^2}\right)-\sqrt{a^2-1}\arctan \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\frac{1}{2}\ln \left(\frac{x^2+{\left(y\left(x\right)\right)}^2}{x^2}\right)+\sqrt{a^2-1}\arctan \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'implicit')

Maple raw output

ln(x)+1/2*ln((x^2+y(x)^2)/x^2)-(a^2-1)^(1/2)*arctan(y(x)/x)-_C1 = 0, ln(x)+1/2*l
n((x^2+y(x)^2)/x^2)+(a^2-1)^(1/2)*arctan(y(x)/x)-_C1 = 0