4.21.1 $a^2(-x^2)-3a^{2}xy(x)y^{\prime }(x)+(1-a^2)y(x{)}^2{y}^{\prime }(x{)}^2+y(x{)}^2=0$

ODE
$$a^2(-x^2)-3a^{2}xy(x)y^{\prime }(x)+(1-a^2)y(x{)}^2{y}^{\prime }(x{)}^2+y(x{)}^2=0$$ ODE Classification

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

Book solution method
Change of variable

Mathematica ✓
cpu = 2.02438 (sec), leaf count = 763

$$\{\text{Solve}[\frac{\log \left(\frac{a^2{x}^4+(3a^2-1)x^{2}y(x{)}^2+(a^2-1)y(x{)}^4}{x^4}\right)+\frac{2{\tan }^{-1}\left(\frac{(3a^2-1)x^2+2(a^2-1)y(x{)}^2}{\sqrt{-5a^4+2a^2-1}{x}^2}\right)}{\sqrt{-5a^4+2a^2-1}}+\frac{2\sqrt{5a^4-2a^2-2\sqrt{5a^4-2a^2+1}+2}{\tanh }^{-1}\left(\frac{\sqrt{\frac{a^2(5a^2+4)x^2+4(a^2-1)y(x{)}^2}{x^2}}}{\sqrt{5a^4-2a^2-2\sqrt{5a^4-2a^2+1}+2}}\right)}{\sqrt{5a^4-2a^2+1}}-\frac{2\sqrt{5a^4-2a^2+2\sqrt{5a^4-2a^2+1}+2}{\tanh }^{-1}\left(\frac{\sqrt{\frac{a^2(5a^2+4)x^2+4(a^2-1)y(x{)}^2}{x^2}}}{\sqrt{5a^4-2a^2+2\sqrt{5a^4-2a^2+1}+2}}\right)}{\sqrt{5a^4-2a^2+1}}}{8a^2-8}=\frac{\log (-2(a^2-1)x)}{2-2a^2}+c_1,y(x)],\text{Solve}[\frac{\log \left(\frac{a^2{x}^4+(3a^2-1)x^{2}y(x{)}^2+(a^2-1)y(x{)}^4}{x^4}\right)+\frac{2{\tan }^{-1}\left(\frac{(3a^2-1)x^2+2(a^2-1)y(x{)}^2}{\sqrt{-5a^4+2a^2-1}{x}^2}\right)}{\sqrt{-5a^4+2a^2-1}}-\frac{2\sqrt{5a^4-2a^2-2\sqrt{5a^4-2a^2+1}+2}{\tanh }^{-1}\left(\frac{\sqrt{\frac{a^2(5a^2+4)x^2+4(a^2-1)y(x{)}^2}{x^2}}}{\sqrt{5a^4-2a^2-2\sqrt{5a^4-2a^2+1}+2}}\right)}{\sqrt{5a^4-2a^2+1}}+\frac{2\sqrt{5a^4-2a^2+2\sqrt{5a^4-2a^2+1}+2}{\tanh }^{-1}\left(\frac{\sqrt{\frac{a^2(5a^2+4)x^2+4(a^2-1)y(x{)}^2}{x^2}}}{\sqrt{5a^4-2a^2+2\sqrt{5a^4-2a^2+1}+2}}\right)}{\sqrt{5a^4-2a^2+1}}}{8a^2-8}=\frac{\log (-2(a^2-1)x)}{2-2a^2}+c_1,y(x)]\}$$

Maple ✓
cpu = 1.169 (sec), leaf count = 181

$$\{\ln \left(x\right)-\frac{1}{2}{\int }^{\frac{y\left(x\right)}{x}}\frac{1}{(a^2-1){\mathit{\_}\mathit{a}}^4+(3a^2-1){\mathit{\_}\mathit{a}}^2+a^2}(\mathit{\_}\mathit{a}\sqrt{5a^4+(4{\mathit{\_}\mathit{a}}^2+4)a^2-4{\mathit{\_}\mathit{a}}^2}+(-2a^2+2){\mathit{\_}\mathit{a}}^3-3\mathit{\_}\mathit{a}{a}^2)d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\frac{1}{2}{\int }^{\frac{y\left(x\right)}{x}}\frac{\mathit{\_}\mathit{a}}{(a^2-1){\mathit{\_}\mathit{a}}^4+(3a^2-1){\mathit{\_}\mathit{a}}^2+a^2}(2{\mathit{\_}\mathit{a}}^2{a}^2-2{\mathit{\_}\mathit{a}}^2+3a^2+\sqrt{5a^4+(4{\mathit{\_}\mathit{a}}^2+4)a^2-4{\mathit{\_}\mathit{a}}^2})d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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*S
qrt[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]/Sq
rt[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