##### 4.20.40 $$\left (\left (1-4 a^2\right ) x^2+y(x)^2\right ) y'(x)^2-8 a^2 x y(x) y'(x)+\left (1-4 a^2\right ) y(x)^2+x^2=0$$

ODE
$\left (\left (1-4 a^2\right ) x^2+y(x)^2\right ) y'(x)^2-8 a^2 x y(x) y'(x)+\left (1-4 a^2\right ) y(x)^2+x^2=0$ 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 = 0.52413 (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 = 0.198 (sec), leaf count = 173

$\left [y \left (x \right ) = \RootOf \left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_a}^{3}+8 \textit {\_a} \,a^{2}+\sqrt {4 \textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+8 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+4 a^{2}-1}-\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} +\textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}+\sqrt {4 \textit {\_a}^{4} a^{2}-\textit {\_a}^{4}+8 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+4 a^{2}-1}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right )+\textit {\_C1} \right ) x\right ]$ 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 + S
qrt[-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*Sqr
t[-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))

Maple raw output

[y(x) = RootOf(-ln(x)+Intat((-_a^3+8*_a*a^2+(4*_a^4*a^2-_a^4+8*_a^2*a^2-2*_a^2+4
*a^2-1)^(1/2)-_a)/(_a^4-16*_a^2*a^2+2*_a^2+1),_a = _Z)+_C1)*x, y(x) = RootOf(-ln
(x)-Intat((_a^3-8*_a*a^2+(4*_a^4*a^2-_a^4+8*_a^2*a^2-2*_a^2+4*a^2-1)^(1/2)+_a)/(
_a^4-16*_a^2*a^2+2*_a^2+1),_a = _Z)+_C1)*x]