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`], _rational, _dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 1.24937 (sec), leaf count = 309
\[\left \{\text {Solve}\left [\frac {2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 a \log \left (x-a^2 x\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=c_1,y(x)\right ],\text {Solve}\left [\frac {-2 i \tan ^{-1}\left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-a \tanh ^{-1}\left (\frac {-a^2-\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \tanh ^{-1}\left (\frac {-a^2+\frac {i y(x)}{x}+1}{a \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 a \log \left (x-a^2 x\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 15.528 (sec), leaf count = 229
\[\left [\ln \left (x \right )-\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {-\frac {a^{2} x^{2}-x^{2}-y \left (x \right )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}+\frac {\ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{a}-\textit {\_C1} = 0, \ln \left (x \right )+\frac {\sqrt {-a^{2}}\, \arctan \left (\frac {a^{2} y \left (x \right )}{\sqrt {-a^{2}}\, \sqrt {-\frac {a^{2} x^{2}-x^{2}-y \left (x \right )^{2}}{x^{2}}}\, x}\right )}{a}+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {\sqrt {\frac {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}{x^{2}}}\, x +y \left (x \right )}{x}\right )}{a}-\textit {\_C1} = 0\right ]\] 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*ArcTanh[(1 - a^2 - (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] - a*ArcTanh[(1 - a^2 + (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + 2*a*Log[x - a^2*x] + a*Log[1 + y[x]^2/x^2])/(-2 + 2*a^2) == C[1], y[x]], Solve[((-2*I)*ArcTan[y[x]/(x*Sqrt[-1 + a^2 - y[x]^2/x^2])] - a*ArcTanh[(1 - a^2 - (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + a*ArcTanh[(1 - a^2 + (I*y[x])/x)/(a*Sqrt[-1 + a^2 - y[x]^2/x^2])] + 2*a*Log[x - a^2*x] + a*Log[1 + y[x]^2/x^2])/(-2 + 2*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))
Maple raw output
[ln(x)-1/a*(-a^2)^(1/2)*arctan(a^2/(-a^2)^(1/2)/(-(a^2*x^2-x^2-y(x)^2)/x^2)^(1/2)*y(x)/x)+1/2*ln((x^2+y(x)^2)/x^2)+1/a*ln((((-a^2*x^2+x^2+y(x)^2)/x^2)^(1/2)*x+y(x))/x)-_C1 = 0, ln(x)+1/a*(-a^2)^(1/2)*arctan(a^2/(-a^2)^(1/2)/(-(a^2*x^2-x^2-y(x)^2)/x^2)^(1/2)*y(x)/x)+1/2*ln((x^2+y(x)^2)/x^2)-1/a*ln((((-a^2*x^2+x^2+y(x)^2)/x^2)^(1/2)*x+y(x))/x)-_C1 = 0]