4.14.7 \(\left (3 x y(x)^2-4 x+1\right ) y'(x)=y(x) \left (2-y(x)^2\right )\)

ODE
\[ \left (3 x y(x)^2-4 x+1\right ) y'(x)=y(x) \left (2-y(x)^2\right ) \] ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.159935 (sec), leaf count = 2348

\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3+2 \left (2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}+\sqrt [3]{2}\right ) x^2-4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x+\left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {8 \sqrt [3]{2} x^4+8 \sqrt [3]{2} x^3+2 \left (2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}+\sqrt [3]{2}\right ) x^2-4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x+\left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{\sqrt {6}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {8 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) x^4+8 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) x^3+2 \left (4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}+i \sqrt [3]{2} \sqrt {3}-\sqrt [3]{2}\right ) x^2-8 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x-i \left (-i+\sqrt {3}\right ) \left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {8 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) x^4+8 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) x^3+2 \left (4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}+i \sqrt [3]{2} \sqrt {3}-\sqrt [3]{2}\right ) x^2-8 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x-i \left (-i+\sqrt {3}\right ) \left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {-8 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) x^4-8 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) x^3+2 \left (4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}-i \sqrt [3]{2} \sqrt {3}-\sqrt [3]{2}\right ) x^2-8 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x+i \left (i+\sqrt {3}\right ) \left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{2 \sqrt {3}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {-8 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) x^4-8 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) x^3+2 \left (4 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}-i \sqrt [3]{2} \sqrt {3}-\sqrt [3]{2}\right ) x^2-8 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}} x+i \left (i+\sqrt {3}\right ) \left (32 x^6+48 x^5-6 \left (9 c_1^2-4\right ) x^4+4 x^3+6 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}\right ){}^{2/3}}{x^2 \sqrt [3]{16 x^6+24 x^5-3 \left (9 c_1^2-4\right ) x^4+2 x^3+3 \sqrt {3} \sqrt {-x^7 c_1^2 \left (32 x^3+48 x^2-27 c_1^2 x+24 x+4\right )}}}}}{2 \sqrt {3}}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 24

\[ \left \{ x+ \left (y \relax (x ) \right ) ^{-2}-{\frac {{\it \_C1}}{ \left (y \relax (x ) \right ) ^{2}}{\frac {1}{\sqrt { \left (y \relax (x ) \right ) ^{2}-2}}}}=0 \right \} \] Mathematica raw input

DSolve[(1 - 4*x + 3*x*y[x]^2)*y'[x] == y[x]*(2 - y[x]^2),y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[(8*2^(1/3)*x^3 + 8*2^(1/3)*x^4 - 4*x*(2*x^3 + 24*x^5 + 16*x^6 -
 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3
 - 27*x*C[1]^2))])^(1/3) + (4*x^3 + 48*x^5 + 32*x^6 - 6*x^4*(-4 + 9*C[1]^2) + 6*
Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(2/3) + 
2*x^2*(2^(1/3) + 2*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*
Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3)))/(x^2*(2*
x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 +
 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3))]/Sqrt[6])}, {y[x] -> Sqrt[(8*2^
(1/3)*x^3 + 8*2^(1/3)*x^4 - 4*x*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2)
 + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/
3) + (4*x^3 + 48*x^5 + 32*x^6 - 6*x^4*(-4 + 9*C[1]^2) + 6*Sqrt[3]*Sqrt[-(x^7*C[1
]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(2/3) + 2*x^2*(2^(1/3) + 2*(2*
x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 +
 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3)))/(x^2*(2*x^3 + 24*x^5 + 16*x^6 
- 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^
3 - 27*x*C[1]^2))])^(1/3))]/Sqrt[6]}, {y[x] -> -Sqrt[((8*I)*2^(1/3)*(I + Sqrt[3]
)*x^3 + (8*I)*2^(1/3)*(I + Sqrt[3])*x^4 - 8*x*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(
-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*
C[1]^2))])^(1/3) - I*(-I + Sqrt[3])*(4*x^3 + 48*x^5 + 32*x^6 - 6*x^4*(-4 + 9*C[1
]^2) + 6*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])
^(2/3) + 2*x^2*(-2^(1/3) + I*2^(1/3)*Sqrt[3] + 4*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^
4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27
*x*C[1]^2))])^(1/3)))/(x^2*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*
Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3))]/
(2*Sqrt[3])}, {y[x] -> Sqrt[((8*I)*2^(1/3)*(I + Sqrt[3])*x^3 + (8*I)*2^(1/3)*(I 
+ Sqrt[3])*x^4 - 8*x*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3
]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3) - I*(-I 
+ Sqrt[3])*(4*x^3 + 48*x^5 + 32*x^6 - 6*x^4*(-4 + 9*C[1]^2) + 6*Sqrt[3]*Sqrt[-(x
^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(2/3) + 2*x^2*(-2^(1/3) 
+ I*2^(1/3)*Sqrt[3] + 4*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqr
t[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3)))/(x^
2*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2
*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3))]/(2*Sqrt[3])}, {y[x] -> -S
qrt[((-8*I)*2^(1/3)*(-I + Sqrt[3])*x^3 - (8*I)*2^(1/3)*(-I + Sqrt[3])*x^4 - 8*x*
(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(
4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3) + I*(I + Sqrt[3])*(4*x^3 + 48
*x^5 + 32*x^6 - 6*x^4*(-4 + 9*C[1]^2) + 6*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 
48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(2/3) + 2*x^2*(-2^(1/3) - I*2^(1/3)*Sqrt[3] + 
4*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2
*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))])^(1/3)))/(x^2*(2*x^3 + 24*x^5 + 16
*x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 
32*x^3 - 27*x*C[1]^2))])^(1/3))]/(2*Sqrt[3])}, {y[x] -> Sqrt[((-8*I)*2^(1/3)*(-I
 + Sqrt[3])*x^3 - (8*I)*2^(1/3)*(-I + Sqrt[3])*x^4 - 8*x*(2*x^3 + 24*x^5 + 16*x^
6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*
x^3 - 27*x*C[1]^2))])^(1/3) + I*(I + Sqrt[3])*(4*x^3 + 48*x^5 + 32*x^6 - 6*x^4*(
-4 + 9*C[1]^2) + 6*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*
C[1]^2))])^(2/3) + 2*x^2*(-2^(1/3) - I*2^(1/3)*Sqrt[3] + 4*(2*x^3 + 24*x^5 + 16*
x^6 - 3*x^4*(-4 + 9*C[1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 3
2*x^3 - 27*x*C[1]^2))])^(1/3)))/(x^2*(2*x^3 + 24*x^5 + 16*x^6 - 3*x^4*(-4 + 9*C[
1]^2) + 3*Sqrt[3]*Sqrt[-(x^7*C[1]^2*(4 + 24*x + 48*x^2 + 32*x^3 - 27*x*C[1]^2))]
)^(1/3))]/(2*Sqrt[3])}}

Maple raw input

dsolve((1-4*x+3*x*y(x)^2)*diff(y(x),x) = (2-y(x)^2)*y(x), y(x),'implicit')

Maple raw output

x+1/y(x)^2-1/(y(x)^2-2)^(1/2)/y(x)^2*_C1 = 0