ODE
\[ a^2 \left (-x^2\right )-3 a^2 x y(x) y'(x)+\left (1-a^2\right ) y(x)^2 y'(x)^2+y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Change of variable
Mathematica ✓
cpu = 2.02438 (sec), leaf count = 763
\[\left \{\text {Solve}\left [\frac {\log \left (\frac {a^2 x^4+\left (3 a^2-1\right ) x^2 y(x)^2+\left (a^2-1\right ) y(x)^4}{x^4}\right )+\frac {2 \tan ^{-1}\left (\frac {\left (3 a^2-1\right ) x^2+2 \left (a^2-1\right ) y(x)^2}{\sqrt {-5 a^4+2 a^2-1} x^2}\right )}{\sqrt {-5 a^4+2 a^2-1}}+\frac {2 \sqrt {5 a^4-2 a^2-2 \sqrt {5 a^4-2 a^2+1}+2} \tanh ^{-1}\left (\frac {\sqrt {\frac {a^2 \left (5 a^2+4\right ) x^2+4 \left (a^2-1\right ) y(x)^2}{x^2}}}{\sqrt {5 a^4-2 a^2-2 \sqrt {5 a^4-2 a^2+1}+2}}\right )}{\sqrt {5 a^4-2 a^2+1}}-\frac {2 \sqrt {5 a^4-2 a^2+2 \sqrt {5 a^4-2 a^2+1}+2} \tanh ^{-1}\left (\frac {\sqrt {\frac {a^2 \left (5 a^2+4\right ) x^2+4 \left (a^2-1\right ) y(x)^2}{x^2}}}{\sqrt {5 a^4-2 a^2+2 \sqrt {5 a^4-2 a^2+1}+2}}\right )}{\sqrt {5 a^4-2 a^2+1}}}{8 a^2-8}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ],\text {Solve}\left [\frac {\log \left (\frac {a^2 x^4+\left (3 a^2-1\right ) x^2 y(x)^2+\left (a^2-1\right ) y(x)^4}{x^4}\right )+\frac {2 \tan ^{-1}\left (\frac {\left (3 a^2-1\right ) x^2+2 \left (a^2-1\right ) y(x)^2}{\sqrt {-5 a^4+2 a^2-1} x^2}\right )}{\sqrt {-5 a^4+2 a^2-1}}-\frac {2 \sqrt {5 a^4-2 a^2-2 \sqrt {5 a^4-2 a^2+1}+2} \tanh ^{-1}\left (\frac {\sqrt {\frac {a^2 \left (5 a^2+4\right ) x^2+4 \left (a^2-1\right ) y(x)^2}{x^2}}}{\sqrt {5 a^4-2 a^2-2 \sqrt {5 a^4-2 a^2+1}+2}}\right )}{\sqrt {5 a^4-2 a^2+1}}+\frac {2 \sqrt {5 a^4-2 a^2+2 \sqrt {5 a^4-2 a^2+1}+2} \tanh ^{-1}\left (\frac {\sqrt {\frac {a^2 \left (5 a^2+4\right ) x^2+4 \left (a^2-1\right ) y(x)^2}{x^2}}}{\sqrt {5 a^4-2 a^2+2 \sqrt {5 a^4-2 a^2+1}+2}}\right )}{\sqrt {5 a^4-2 a^2+1}}}{8 a^2-8}=\frac {\log \left (-2 \left (a^2-1\right ) x\right )}{2-2 a^2}+c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 1.169 (sec), leaf count = 181
\[ \left \{ \ln \left ( x \right ) -{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{ \left ( {a}^{2}-1 \right ) {{\it \_a}}^{4}+ \left ( 3\,{a}^{2}-1 \right ) {{\it \_a}}^{2}+{a}^{2}} \left ( {\it \_a}\,\sqrt {5\,{a}^{4}+ \left ( 4\,{{\it \_a}}^{2}+4 \right ) {a}^{2}-4\,{{\it \_a}}^{2}}+ \left ( -2\,{a}^{2}+2 \right ) {{\it \_a}}^{3}-3\,{\it \_a}\,{a}^{2} \right ) }{d{\it \_a}}}-{\it \_C1}=0,\ln \left ( x \right ) +{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {{\it \_a}}{ \left ( {a}^{2}-1 \right ) {{\it \_a}}^{4}+ \left ( 3\,{a}^{2}-1 \right ) {{\it \_a}}^{2}+{a}^{2}} \left ( 2\,{{\it \_a}}^{2}{a}^{2}-2\,{{\it \_a}}^{2}+3\,{a}^{2}+\sqrt {5\,{a}^{4}+ \left ( 4\,{{\it \_a}}^{2}+4 \right ) {a}^{2}-4\,{{\it \_a}}^{2}} \right ) }{d{\it \_a}}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[-(a^2*x^2) + y[x]^2 - 3*a^2*x*y[x]*y'[x] + (1 - a^2)*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[((2*ArcTan[((-1 + 3*a^2)*x^2 + 2*(-1 + a^2)*y[x]^2)/(Sqrt[-1 + 2*a^2 - 5*a^4]*x^2)])/Sqrt[-1 + 2*a^2 - 5*a^4] + (2*Sqrt[2 - 2*a^2 + 5*a^4 - 2*Sqrt[1 - 2*a^2 + 5*a^4]]*ArcTanh[Sqrt[(a^2*(4 + 5*a^2)*x^2 + 4*(-1 + a^2)*y[x]^2)/x^2]/Sqrt[2 - 2*a^2 + 5*a^4 - 2*Sqrt[1 - 2*a^2 + 5*a^4]]])/Sqrt[1 - 2*a^2 + 5*a^4] - (2*Sqrt[2 - 2*a^2 + 5*a^4 + 2*Sqrt[1 - 2*a^2 + 5*a^4]]*ArcTanh[Sqrt[(a^2*(4 + 5*a^2)*x^2 + 4*(-1 + a^2)*y[x]^2)/x^2]/Sqrt[2 - 2*a^2 + 5*a^4 + 2*Sqrt[1 - 2*a^2 + 5*a^4]]])/Sqrt[1 - 2*a^2 + 5*a^4] + Log[(a^2*x^4 + (-1 + 3*a^2)*x^2*y[x]^2 + (-1 + a^2)*y[x]^4)/x^4])/(-8 + 8*a^2) == C[1] + Log[-2*(-1 + a^2)*x]/(2 - 2*a^2), y[x]], Solve[((2*ArcTan[((-1 + 3*a^2)*x^2 + 2*(-1 + a^2)*y[x]^2)/(Sqrt[-1 + 2*a^2 - 5*a^4]*x^2)])/Sqrt[-1 + 2*a^2 - 5*a^4] - (2*Sqrt[2 - 2*a^2 + 5*a^4 - 2*Sqrt[1 - 2*a^2 + 5*a^4]]*ArcTanh[Sqrt[(a^2*(4 + 5*a^2)*x^2 + 4*(-1 + a^2)*y[x]^2)/x^2]/Sqrt[2 - 2*a^2 + 5*a^4 - 2*Sqrt[1 - 2*a^2 + 5*a^4]]])/Sqrt[1 - 2*a^2 + 5*a^4] + (2*Sqrt[2 - 2*a^2 + 5*a^4 + 2*Sqrt[1 - 2*a^2 + 5*a^4]]*ArcTanh[Sqrt[(a^2*(4 + 5*a^2)*x^2 + 4*(-1 + a^2)*y[x]^2)/x^2]/Sqrt[2 - 2*a^2 + 5*a^4 + 2*Sqrt[1 - 2*a^2 + 5*a^4]]])/Sqrt[1 - 2*a^2 + 5*a^4] + Log[(a^2*x^4 + (-1 + 3*a^2)*x^2*y[x]^2 + (-1 + a^2)*y[x]^4)/x^4])/(-8 + 8*a^2) == C[1] + Log[-2*(-1 + a^2)*x]/(2 - 2*a^2), y[x]]}
Maple raw input
dsolve((-a^2+1)*y(x)^2*diff(y(x),x)^2-3*a^2*x*y(x)*diff(y(x),x)-a^2*x^2+y(x)^2 = 0, y(x),'implicit')
Maple raw output
ln(x)+1/2*Intat((2*_a^2*a^2-2*_a^2+3*a^2+(5*a^4+(4*_a^2+4)*a^2-4*_a^2)^(1/2))*_a/((a^2-1)*_a^4+(3*a^2-1)*_a^2+a^2),_a = y(x)/x)-_C1 = 0, ln(x)-1/2*Intat((_a*(5*a^4+(4*_a^2+4)*a^2-4*_a^2)^(1/2)+(-2*a^2+2)*_a^3-3*_a*a^2)/((a^2-1)*_a^4+(3*a^2-1)*_a^2+a^2),_a = y(x)/x)-_C1 = 0