4.20.40 $((1-4a^2)x^2+y(x{)}^2)y^{\prime }(x{)}^2-8a^{2}xy(x)y^{\prime }(x)+(1-4a^2)y(x{)}^2+x^2=0$

ODE
$$((1-4a^2)x^2+y(x{)}^2)y^{\prime }(x{)}^2-8a^{2}xy(x)y^{\prime }(x)+(1-4a^2)y(x{)}^2+x^2=0$$ ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous ODE, $x^{n}f(\frac{y}{x},y^{\prime })=0$, Solve for $p$

Mathematica ✓
cpu = 0.391856 (sec), leaf count = 317

$$\{\text{Solve}[c_1=\text{RootSum}[-{\mathrm{\#}\text{1}}^3+{\mathrm{\#}\text{1}}^2\sqrt{2a-1}\sqrt{2a+1}+8\mathrm{\#}\text{1}{a}^2-\mathrm{\#}\text{1}+\sqrt{2a-1}\sqrt{2a+1}\mathrm{\&},\frac{-{\mathrm{\#}\text{1}}^2\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})+4a^2\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})-\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})}{-3{\mathrm{\#}\text{1}}^2+2\mathrm{\#}\text{1}\sqrt{2a-1}\sqrt{2a+1}+8a^2-1}\mathrm{\&}]+\log (x),y(x)],\text{Solve}[c_1=\text{RootSum}[{\mathrm{\#}\text{1}}^3+{\mathrm{\#}\text{1}}^2\sqrt{2a-1}\sqrt{2a+1}-8\mathrm{\#}\text{1}{a}^2+\mathrm{\#}\text{1}+\sqrt{2a-1}\sqrt{2a+1}\mathrm{\&},\frac{-{\mathrm{\#}\text{1}}^2\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})+4a^2\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})-\log (\frac{y(x)}{x}-\mathrm{\#}\text{1})}{-3{\mathrm{\#}\text{1}}^2-2\mathrm{\#}\text{1}\sqrt{2a-1}\sqrt{2a+1}+8a^2-1}\mathrm{\&}]+\log (x),y(x)]\}$$

Maple ✓
cpu = 0.544 (sec), leaf count = 137

$$\{\ln \left(x\right)+{\int }^{\frac{y\left(x\right)}{x}}\frac{1}{{\mathit{\_}\mathit{a}}^4-16{\mathit{\_}\mathit{a}}^2{a}^2+2{\mathit{\_}\mathit{a}}^2+1}({\mathit{\_}\mathit{a}}^3-8\mathit{\_}\mathit{a}{a}^2+\sqrt{{({\mathit{\_}\mathit{a}}^2+1)}^2(4a^2-1)}+\mathit{\_}\mathit{a})d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)-{\int }^{\frac{y\left(x\right)}{x}}\frac{1}{-1-{\mathit{\_}\mathit{a}}^4+(16a^2-2){\mathit{\_}\mathit{a}}^2}(-8\mathit{\_}\mathit{a}{a}^2+{\mathit{\_}\mathit{a}}^3+\mathit{\_}\mathit{a}-\sqrt{{({\mathit{\_}\mathit{a}}^2+1)}^2(4a^2-1)})d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'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