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 = 1.2119 (sec), leaf count = 311
\[\left \{\text {Solve}\left [\frac {-\log \left (\frac {2 \left (a^2-1\right ) y(x)^2}{x^2}-\sqrt {\frac {4 \left (a^2-1\right ) y(x)^2}{x^2}+\left (5 a^2+4\right ) a^2}+3 a^2\right )-2 \log \left (-2 \left (a^2-1\right ) x\right )+4 \left (a^2-1\right ) c_1+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\frac {4 \left (a^2-1\right ) y(x)^2}{x^2}+\left (5 a^2+4\right ) a^2}-1}{\sqrt {5 a^4-2 a^2+1}}\right )}{\sqrt {5 a^4-2 a^2+1}}}{a^2-1}=0,y(x)\right ],\text {Solve}\left [\frac {\log \left (\frac {2 \left (a^2-1\right ) y(x)^2}{x^2}+\sqrt {\frac {4 \left (a^2-1\right ) y(x)^2}{x^2}+\left (5 a^2+4\right ) a^2}+3 a^2\right )+2 \log \left (-2 \left (a^2-1\right ) x\right )-4 \left (a^2-1\right ) c_1+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\frac {4 \left (a^2-1\right ) y(x)^2}{x^2}+\left (5 a^2+4\right ) a^2}+1}{\sqrt {5 a^4-2 a^2+1}}\right )}{\sqrt {5 a^4-2 a^2+1}}}{a^2-1}=0,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.43 (sec), leaf count = 195
\[\left [y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} \right )+2 \textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\left (-2 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}-3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{\textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 \textit {\_C1} \right ) x\right ]\] 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*ArcTanh[(-1 + Sqrt[a^2*(4 + 5*a^2) + (4*(-1 + a^2)*y[x]^2)/x^2])/Sqrt[1 - 2*a^2 + 5*a^4]])/Sqrt[1 - 2*a^2 + 5*a^4] + 4*(-1 + a^2)*C[1] - 2*Log[-2*(-1 + a^2)*x] - Log[3*a^2 + (2*(-1 + a^2)*y[x]^2)/x^2 - Sqrt[a^2*(4 + 5*a^2) + (4*(-1 + a^2)*y[x]^2)/x^2]])/(-1 + a^2) == 0, y[x]], Solve[((2*ArcTanh[(1 + Sqrt[a^2*(4 + 5*a^2) + (4*(-1 + a^2)*y[x]^2)/x^2])/Sqrt[1 - 2*a^2 + 5*a^4]])/Sqrt[1 - 2*a^2 + 5*a^4] - 4*(-1 + a^2)*C[1] + 2*Log[-2*(-1 + a^2)*x] + Log[3*a^2 + (2*(-1 + a^2)*y[x]^2)/x^2 + Sqrt[a^2*(4 + 5*a^2) + (4*(-1 + a^2)*y[x]^2)/x^2]])/(-1 + a^2) == 0, 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))
Maple raw output
[y(x) = RootOf(-2*ln(x)-Intat((2*_a^2*a^2-2*_a^2+3*a^2+(4*_a^2*a^2+5*a^4-4*_a^2+4*a^2)^(1/2))/(_a^4*a^2-_a^4+3*_a^2*a^2-_a^2+a^2)*_a,_a = _Z)+2*_C1)*x, y(x) = RootOf(-2*ln(x)+Intat(1/(_a^4*a^2-_a^4+3*_a^2*a^2-_a^2+a^2)*(-2*_a^2*a^2+2*_a^2-3*a^2+(4*_a^2*a^2+5*a^4-4*_a^2+4*a^2)^(1/2))*_a,_a = _Z)+2*_C1)*x]