4.21.4 $a^{2}x+y(x{)}^3(-y^{\prime }(x))+xy(x{)}^2{y}^{\prime }(x{)}^2=0$

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

[[_homogeneous, `class G`], _rational]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.345739 (sec), leaf count = 132

$$\{\{y(x)\to -\sqrt{2a^2{e}^{c_1}+\frac{1}{2}{e}^{-c_1}{x}^2}\},\{y(x)\to \sqrt{2a^2{e}^{c_1}+\frac{1}{2}{e}^{-c_1}{x}^2}\},\{y(x)\to -\frac{\sqrt{4a^2{e}^{-c_1}{x}^2+e^{c_1}}}{\sqrt{2}}\},\{y(x)\to \frac{\sqrt{4a^2{e}^{-c_1}{x}^2+e^{c_1}}}{\sqrt{2}}\}\}$$

Maple ✓
cpu = 1.575 (sec), leaf count = 161

$$\{1(\ln \left(\frac{1}{x}(2{\left(y\left(x\right)\right)}^4+2\sqrt{{\left(y\left(x\right)\right)}^4}\sqrt{{\left(y\left(x\right)\right)}^4-4a^2{x}^2})\right){\left(y\left(x\right)\right)}^2+(\mathit{\_}\mathit{C}\mathit{1}-\ln \left(x\right))\sqrt{{\left(y\left(x\right)\right)}^4}-2{\left(y\left(x\right)\right)}^2\ln \left(y\left(x\right)\right))\frac{1}{\sqrt{{\left(y\left(x\right)\right)}^4}}=0,-1(\ln \left(\frac{1}{x}(2{\left(y\left(x\right)\right)}^4+2\sqrt{{\left(y\left(x\right)\right)}^4}\sqrt{{\left(y\left(x\right)\right)}^4-4a^2{x}^2})\right){\left(y\left(x\right)\right)}^2+(-\mathit{\_}\mathit{C}\mathit{1}+\ln \left(x\right))\sqrt{{\left(y\left(x\right)\right)}^4}-2{\left(y\left(x\right)\right)}^2\ln \left(y\left(x\right)\right))\frac{1}{\sqrt{{\left(y\left(x\right)\right)}^4}}=0,{\left(y\left(x\right)\right)}^4-4a^2{x}^2=0\}$$ Mathematica raw input

DSolve[a^2*x - y[x]^3*y'[x] + x*y[x]^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> -Sqrt[2*a^2*E^C[1] + x^2/(2*E^C[1])]}, {y[x] -> Sqrt[2*a^2*E^C[1] + x^
2/(2*E^C[1])]}, {y[x] -> -(Sqrt[E^C[1] + (4*a^2*x^2)/E^C[1]]/Sqrt[2])}, {y[x] ->
 Sqrt[E^C[1] + (4*a^2*x^2)/E^C[1]]/Sqrt[2]}}

Maple raw input

dsolve(x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)+a^2*x = 0, y(x),'implicit')

Maple raw output

y(x)^4-4*a^2*x^2 = 0, -(ln((2*y(x)^4+2*(y(x)^4)^(1/2)*(y(x)^4-4*a^2*x^2)^(1/2))/
x)*y(x)^2+(-_C1+ln(x))*(y(x)^4)^(1/2)-2*y(x)^2*ln(y(x)))/(y(x)^4)^(1/2) = 0, (ln
((2*y(x)^4+2*(y(x)^4)^(1/2)*(y(x)^4-4*a^2*x^2)^(1/2))/x)*y(x)^2+(_C1-ln(x))*(y(x
)^4)^(1/2)-2*y(x)^2*ln(y(x)))/(y(x)^4)^(1/2) = 0