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

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

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

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.688405 (sec), leaf count = 369

\[\left \{\text {Solve}\left [\frac {a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)+i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )+i y(x)\right )}{a^3 (x-i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ],\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \left (2 \log \left (x-a^2 x\right )-\log \left (\frac {\left (a^2-1\right ) \left (y(x)-i x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )\right )}{a^3 (x-i y(x))}\right )+\log \left (-\frac {i \left (a^2-1\right ) \left (x \left (a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}+a^2-1\right )-i y(x)\right )}{a^3 (x+i y(x))}\right )+\log \left (\frac {y(x)^2}{x^2}+1\right )\right )}{2 \left (a^2-1\right )}=c_1,y(x)\right ]\right \}\]

Maple
cpu = 0.119 (sec), leaf count = 229

\[ \left \{ {\frac {1}{2\,a} \left (-2\,{\it \_C1}\,a+2\,a\ln \relax (x ) +\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) a-2\,\sqrt {-{a}^{2}}\arctan \left ({\frac {{a}^{2}y \relax (x ) }{\sqrt {-{a}^{2}}x}{\frac {1}{\sqrt {{\frac { \left (-{a}^{2}+1 \right ) {x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) +2\,\ln \left ({\frac {1}{x} \left (\sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}x+y \relax (x ) \right ) } \right ) \right ) }=0,{\frac {1}{2\,a} \left (-2\,{\it \_C1}\,a+2\,a\ln \relax (x ) +\ln \left ({\frac {{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) a+2\,\sqrt {-{a}^{2}}\arctan \left ({\frac {{a}^{2}y \relax (x ) }{\sqrt {-{a}^{2}}x}{\frac {1}{\sqrt {{\frac { \left (-{a}^{2}+1 \right ) {x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}}}} \right ) -2\,\ln \left ({\frac {1}{x} \left (\sqrt {{\frac {-{a}^{2}{x}^{2}+{x}^{2}+ \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}x+y \relax (x ) \right ) } \right ) \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{Solve[((-2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*(2*Log[x - a^2*x
] + Log[1 + y[x]^2/x^2] - Log[((-1 + a^2)*(y[x] + I*x*(-1 + a^2 + a*Sqrt[-1 + a^
2 - y[x]^2/x^2])))/(a^3*(x + I*y[x]))] + Log[(I*(-1 + a^2)*(I*y[x] + x*(-1 + a^2
 + a*Sqrt[-1 + a^2 - y[x]^2/x^2])))/(a^3*(x - I*y[x]))]))/(2*(-1 + a^2)) == C[1]
, y[x]], Solve[((2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*(2*Log[x 
- a^2*x] + Log[1 + y[x]^2/x^2] - Log[((-1 + a^2)*(y[x] - I*x*(-1 + a^2 + a*Sqrt[
-1 + a^2 - y[x]^2/x^2])))/(a^3*(x - I*y[x]))] + Log[((-I)*(-1 + a^2)*((-I)*y[x] 
+ x*(-1 + a^2 + a*Sqrt[-1 + a^2 - y[x]^2/x^2])))/(a^3*(x + I*y[x]))]))/(2*(-1 + 
a^2)) == C[1], y[x]]}

Maple raw input

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

Maple raw output

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