ODE
\[ \left (\left (1-4 a^2\right ) x^2+y(x)^2\right ) y'(x)^2-8 a^2 x y(x) y'(x)+\left (1-4 a^2\right ) y(x)^2+x^2=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.391856 (sec), leaf count = 317
\[\left \{\text {Solve}\left [c_1=\text {RootSum}\left [-\text {$\#$1}^3+\text {$\#$1}^2 \sqrt {2 a-1} \sqrt {2 a+1}+8 \text {$\#$1} a^2-\text {$\#$1}+\sqrt {2 a-1} \sqrt {2 a+1}\& ,\frac {-\text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+4 a^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{-3 \text {$\#$1}^2+2 \text {$\#$1} \sqrt {2 a-1} \sqrt {2 a+1}+8 a^2-1}\& \right ]+\log (x),y(x)\right ],\text {Solve}\left [c_1=\text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2 \sqrt {2 a-1} \sqrt {2 a+1}-8 \text {$\#$1} a^2+\text {$\#$1}+\sqrt {2 a-1} \sqrt {2 a+1}\& ,\frac {-\text {$\#$1}^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )+4 a^2 \log \left (\frac {y(x)}{x}-\text {$\#$1}\right )-\log \left (\frac {y(x)}{x}-\text {$\#$1}\right )}{-3 \text {$\#$1}^2-2 \text {$\#$1} \sqrt {2 a-1} \sqrt {2 a+1}+8 a^2-1}\& \right ]+\log (x),y(x)\right ]\right \}\]
Maple ✓
cpu = 0.544 (sec), leaf count = 137
\[ \left \{ \ln \left ( x \right ) +\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{{\it \_a}}^{4}-16\,{{\it \_a}}^{2}{a}^{2}+2\,{{\it \_a}}^{2}+1} \left ( {{\it \_a}}^{3}-8\,{\it \_a}\,{a}^{2}+\sqrt { \left ( {{\it \_a}}^{2}+1 \right ) ^{2} \left ( 4\,{a}^{2}-1 \right ) }+{\it \_a} \right ) }{d{\it \_a}}-{\it \_C1}=0,\ln \left ( x \right ) -\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{-1-{{\it \_a}}^{4}+ \left ( 16\,{a}^{2}-2 \right ) {{\it \_a}}^{2}} \left ( -8\,{\it \_a}\,{a}^{2}+{{\it \_a}}^{3}+{\it \_a}-\sqrt { \left ( {{\it \_a}}^{2}+1 \right ) ^{2} \left ( 4\,{a}^{2}-1 \right ) } \right ) }{d{\it \_a}}-{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x^2 + (1 - 4*a^2)*y[x]^2 - 8*a^2*x*y[x]*y'[x] + ((1 - 4*a^2)*x^2 + y[x]^2)*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{Solve[C[1] == Log[x] + RootSum[Sqrt[-1 + 2*a]*Sqrt[1 + 2*a] - #1 + 8*a^2*#1 + Sqrt[-1 + 2*a]*Sqrt[1 + 2*a]*#1^2 - #1^3 & , (-Log[-#1 + y[x]/x] + 4*a^2*Log[-#1 + y[x]/x] - Log[-#1 + y[x]/x]*#1^2)/(-1 + 8*a^2 + 2*Sqrt[-1 + 2*a]*Sqrt[1 + 2*a]*#1 - 3*#1^2) & ], y[x]], Solve[C[1] == Log[x] + RootSum[Sqrt[-1 + 2*a]*Sqrt[1 + 2*a] + #1 - 8*a^2*#1 + Sqrt[-1 + 2*a]*Sqrt[1 + 2*a]*#1^2 + #1^3 & , (-Log[-#1 + y[x]/x] + 4*a^2*Log[-#1 + y[x]/x] - Log[-#1 + y[x]/x]*#1^2)/(-1 + 8*a^2 - 2*Sqrt[-1 + 2*a]*Sqrt[1 + 2*a]*#1 - 3*#1^2) & ], y[x]]}
Maple raw input
dsolve(((-4*a^2+1)*x^2+y(x)^2)*diff(y(x),x)^2-8*a^2*x*y(x)*diff(y(x),x)+x^2+(-4*a^2+1)*y(x)^2 = 0, y(x),'implicit')
Maple raw output
ln(x)-Intat((-8*_a*a^2+_a^3+_a-((_a^2+1)^2*(4*a^2-1))^(1/2))/(-1-_a^4+(16*a^2-2)*_a^2),_a = y(x)/x)-_C1 = 0, ln(x)+Intat((_a^3-8*_a*a^2+((_a^2+1)^2*(4*a^2-1))^(1/2)+_a)/(_a^4-16*_a^2*a^2+2*_a^2+1),_a = y(x)/x)-_C1 = 0