4.20.26 $y(x{)}^2{y}^{\prime }(x{)}^2-3x{y}^{\prime }(x)+y(x)=0$

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

[[_1st_order, _with_linear_symmetries], _rational]

Book solution method
No Missing Variables ODE, Solve for $x$

Mathematica ✓
cpu = 0.167357 (sec), leaf count = 177

$$\{\{y(x)\to -\sqrt[3]{-2}\sqrt[3]{e^{c_1}(3x-2e^{c_1})}\},\{y(x)\to \sqrt[3]{6e^{c_1}x-4e^{2c_1}}\},\{y(x)\to (-1{)}^{2/3}\sqrt[3]{6e^{c_1}x-4e^{2c_1}}\},\{y(x)\to \frac{\sqrt[3]{-e^{c_1}(e^{c_1}-6x)}}{2^{2/3}}\},\{y(x)\to -\frac{\sqrt[3]{-1}\sqrt[3]{-e^{c_1}(e^{c_1}-6x)}}{2^{2/3}}\},\{y(x)\to {(-\frac{1}{2})}^{2/3}\sqrt[3]{-e^{c_1}(e^{c_1}-6x)}\}\}$$

Maple ✓
cpu = 14.78 (sec), leaf count = 62

$$\{{\left(y\left(x\right)\right)}^3-\frac{9x^2}{4}=0,\ln \left(x\right)-{\int }^{y\left(x\right)x^{-\frac{2}{3}}}\frac{1}{8{\mathit{\_}\mathit{a}}^4-18\mathit{\_}\mathit{a}}(-12{\mathit{\_}\mathit{a}}^3+9\sqrt{-4{\mathit{\_}\mathit{a}}^3+9}+27)d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((-2)^(1/3)*(E^C[1]*(-2*E^C[1] + 3*x))^(1/3))}, {y[x] -> (-4*E^(2*C[1
]) + 6*E^C[1]*x)^(1/3)}, {y[x] -> (-1)^(2/3)*(-4*E^(2*C[1]) + 6*E^C[1]*x)^(1/3)}
, {y[x] -> (-(E^C[1]*(E^C[1] - 6*x)))^(1/3)/2^(2/3)}, {y[x] -> -(((-1)^(1/3)*(-(
E^C[1]*(E^C[1] - 6*x)))^(1/3))/2^(2/3))}, {y[x] -> (-1/2)^(2/3)*(-(E^C[1]*(E^C[1
] - 6*x)))^(1/3)}}

Maple raw input

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

Maple raw output

y(x)^3-9/4*x^2 = 0, ln(x)-Intat((-12*_a^3+9*(-4*_a^3+9)^(1/2)+27)/(8*_a^4-18*_a)
,_a = y(x)/x^(2/3))-_C1 = 0