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 = 0.176591 (sec), leaf count = 93
\[\left \{\text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \tan ^{-1}\left (\frac {y(x)}{x}\right )=c_1+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )+\log (x),y(x)\right ],\text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \tan ^{-1}\left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )+\log (x)=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.545 (sec), leaf count = 78
\[ \left \{ \ln \left ( x \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }-\sqrt {{a}^{2}-1}\arctan \left ( {\frac {y \left ( x \right ) }{x}} \right ) -{\it \_C1}=0,\ln \left ( x \right ) +{\frac {1}{2}\ln \left ( {\frac {{x}^{2}+ \left ( y \left ( x \right ) \right ) ^{2}}{{x}^{2}}} \right ) }+\sqrt {{a}^{2}-1}\arctan \left ( {\frac {y \left ( x \right ) }{x}} \right ) -{\it \_C1}=0 \right \} \] 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),'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*ln((x^2+y(x)^2)/x^2)+(a^2-1)^(1/2)*arctan(y(x)/x)-_C1 = 0