4.19.34 \(x^3 y'(x)^2-\left (2 x^2 y(x)+1\right ) y'(x)+x y(x)^2=0\)

ODE
\[ x^3 y'(x)^2-\left (2 x^2 y(x)+1\right ) y'(x)+x y(x)^2=0 \] ODE Classification

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

Book solution method
Change of variable

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

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

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

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {1}{4}\ln \left (-1+\sqrt {4\,{x}^{2}y \relax (x ) +1} \right ) }-{\frac {1}{4}\ln \left (\sqrt {4\,{x}^{2}y \relax (x ) +1}+1 \right ) }+{\frac {1}{12}\ln \left (3\,\sqrt {4\,{x}^{2}y \relax (x ) +1}+1 \right ) }-{\frac {1}{12}\ln \left (3\,\sqrt {4\,{x}^{2}y \relax (x ) +1}-1 \right ) }-{\frac {\ln \left (9\,{x}^{2}y \relax (x ) +2 \right ) }{12}}-{\frac {\ln \left ({x}^{2}y \relax (x ) \right ) }{4}}=0,\ln \relax (x ) -{\it \_C1}-{\frac {\ln \left (9\,{x}^{2}y \relax (x ) +2 \right ) }{12}}-{\frac {\ln \left ({x}^{2}y \relax (x ) \right ) }{4}}-{\frac {1}{4}\ln \left (-1+\sqrt {4\,{x}^{2}y \relax (x ) +1} \right ) }+{\frac {1}{4}\ln \left (\sqrt {4\,{x}^{2}y \relax (x ) +1}+1 \right ) }-{\frac {1}{12}\ln \left (3\,\sqrt {4\,{x}^{2}y \relax (x ) +1}+1 \right ) }+{\frac {1}{12}\ln \left (3\,\sqrt {4\,{x}^{2}y \relax (x ) +1}-1 \right ) }=0 \right \} \] 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