4.19.34 $x^3{y}^{\prime }(x{)}^2-(2x^{2}y(x)+1)y^{\prime }(x)+xy(x{)}^2=0$

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

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

Book solution method
Change of variable

Mathematica ✓
cpu = 0.515414 (sec), leaf count = 1921

$$\{\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},1]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},2]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},3]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},4]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},5]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},6]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},7]\},\{y(x)\to \text{Root}[1024x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+81e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},8]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},1]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},2]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},3]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},4]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},5]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},6]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},7]\},\{y(x)\to \text{Root}[x^{12}-576e^{12c_1}{\mathrm{\#}\text{1}}^4{x}^8-2176e^{12c_1}{\mathrm{\#}\text{1}}^3{x}^6+82944e^{24c_1}{\mathrm{\#}\text{1}}^8{x}^4-1536e^{12c_1}{\mathrm{\#}\text{1}}^2{x}^4+36864e^{24c_1}{\mathrm{\#}\text{1}}^7{x}^2-384e^{12c_1}\mathrm{\#}\text{1}{x}^2-32e^{12c_1}+4096e^{24c_1}{\mathrm{\#}\text{1}}^6\mathrm{\&},8]\}\}$$

Maple ✓
cpu = 0.095 (sec), leaf count = 195

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

DSolve[x*y[x]^2 - (1 + 2*x^2*y[x])*y'[x] + x^3*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> Root[-32*E^(12*C[1]) + 1024*x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12
*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24
*C[1])*#1^6 + 36*E^(24*C[1])*x^2*#1^7 + 81*E^(24*C[1])*x^4*#1^8 & , 1]}, {y[x] -
> Root[-32*E^(12*C[1]) + 1024*x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x
^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#
1^6 + 36*E^(24*C[1])*x^2*#1^7 + 81*E^(24*C[1])*x^4*#1^8 & , 2]}, {y[x] -> Root[-
32*E^(12*C[1]) + 1024*x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 
- 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36
*E^(24*C[1])*x^2*#1^7 + 81*E^(24*C[1])*x^4*#1^8 & , 3]}, {y[x] -> Root[-32*E^(12
*C[1]) + 1024*x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 - 2176*E
^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36*E^(24*C
[1])*x^2*#1^7 + 81*E^(24*C[1])*x^4*#1^8 & , 4]}, {y[x] -> Root[-32*E^(12*C[1]) +
 1024*x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1
])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36*E^(24*C[1])*x^2
*#1^7 + 81*E^(24*C[1])*x^4*#1^8 & , 5]}, {y[x] -> Root[-32*E^(12*C[1]) + 1024*x^
12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#
1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36*E^(24*C[1])*x^2*#1^7 + 
81*E^(24*C[1])*x^4*#1^8 & , 6]}, {y[x] -> Root[-32*E^(12*C[1]) + 1024*x^12 - 384
*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 57
6*E^(12*C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36*E^(24*C[1])*x^2*#1^7 + 81*E^(24
*C[1])*x^4*#1^8 & , 7]}, {y[x] -> Root[-32*E^(12*C[1]) + 1024*x^12 - 384*E^(12*C
[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*
C[1])*x^8*#1^4 + 4*E^(24*C[1])*#1^6 + 36*E^(24*C[1])*x^2*#1^7 + 81*E^(24*C[1])*x
^4*#1^8 & , 8]}, {y[x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 -
 1536*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^
4 + 4096*E^(24*C[1])*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#
1^8 & , 1]}, {y[x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 153
6*E^(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 
4096*E^(24*C[1])*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 
& , 2]}, {y[x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^
(12*C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096
*E^(24*C[1])*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 
3]}, {y[x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*
C[1])*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096*E^(
24*C[1])*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 4]},
 {y[x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1]
)*x^4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096*E^(24*C
[1])*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 5]}, {y[
x] -> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^
4*#1^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096*E^(24*C[1])
*#1^6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 6]}, {y[x] -
> Root[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1
^2 - 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096*E^(24*C[1])*#1^
6 + 36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 7]}, {y[x] -> Ro
ot[-32*E^(12*C[1]) + x^12 - 384*E^(12*C[1])*x^2*#1 - 1536*E^(12*C[1])*x^4*#1^2 -
 2176*E^(12*C[1])*x^6*#1^3 - 576*E^(12*C[1])*x^8*#1^4 + 4096*E^(24*C[1])*#1^6 + 
36864*E^(24*C[1])*x^2*#1^7 + 82944*E^(24*C[1])*x^4*#1^8 & , 8]}}

Maple raw input

dsolve(x^3*diff(y(x),x)^2-(1+2*x^2*y(x))*diff(y(x),x)+x*y(x)^2 = 0, y(x),'implicit')

Maple raw output

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