4.10.7 \((-y(x)-4 x+6) y'(x)=2 x-y(x)\)

ODE
\[ (-y(x)-4 x+6) y'(x)=2 x-y(x) \] ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 0.19625 (sec), leaf count = 2563

\[\left \{\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,1\right ]}+6\right \},\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,2\right ]}+6\right \},\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,3\right ]}+6\right \},\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,4\right ]}+6\right \},\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,5\right ]}+6\right \},\left \{y(x)\to -4 x+\frac {1}{\text {Root}\left [\left (11664 e^{\frac {12 c_1}{25}} x^6-69984 e^{\frac {12 c_1}{25}} x^5+174960 e^{\frac {12 c_1}{25}} x^4+11664 x^4-233280 e^{\frac {12 c_1}{25}} x^3-46656 x^3+174960 e^{\frac {12 c_1}{25}} x^2+69984 x^2-69984 e^{\frac {12 c_1}{25}} x-46656 x+11664 e^{\frac {12 c_1}{25}}+11664\right ) \text {$\#$1}^6+\left (-23328 e^{\frac {12 c_1}{25}} x^5+116640 e^{\frac {12 c_1}{25}} x^4-233280 e^{\frac {12 c_1}{25}} x^3-23328 x^3+233280 e^{\frac {12 c_1}{25}} x^2+69984 x^2-116640 e^{\frac {12 c_1}{25}} x-69984 x+23328 e^{\frac {12 c_1}{25}}+23328\right ) \text {$\#$1}^5+\left (19440 e^{\frac {12 c_1}{25}} x^4-77760 e^{\frac {12 c_1}{25}} x^3+116640 e^{\frac {12 c_1}{25}} x^2+17496 x^2-77760 e^{\frac {12 c_1}{25}} x-34992 x+19440 e^{\frac {12 c_1}{25}}+17496\right ) \text {$\#$1}^4+\left (-8640 e^{\frac {12 c_1}{25}} x^3+25920 e^{\frac {12 c_1}{25}} x^2-25920 e^{\frac {12 c_1}{25}} x-5832 x+8640 e^{\frac {12 c_1}{25}}+5832\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2-4320 e^{\frac {12 c_1}{25}} x+2160 e^{\frac {12 c_1}{25}}+729\right ) \text {$\#$1}^2+\left (288 e^{\frac {12 c_1}{25}}-288 e^{\frac {12 c_1}{25}} x\right ) \text {$\#$1}+16 e^{\frac {12 c_1}{25}}\& ,6\right ]}+6\right \}\right \}\]

Maple
cpu = 0.029 (sec), leaf count = 49

\[ \left \{ -3\,\ln \left ({\frac {3-x-y \relax (x ) }{-1+x}} \right ) +2\,\ln \left ({\frac {-y \relax (x ) +4-2\,x}{-1+x}} \right ) -\ln \left (-1+x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(6 - 4*x - y[x])*y'[x] == 2*x - y[x],y[x],x]

Mathematica raw output

{{y[x] -> 6 - 4*x + Root[16*E^((12*C[1])/25) + (288*E^((12*C[1])/25) - 288*E^((1
2*C[1])/25)*x)*#1 + (729 + 2160*E^((12*C[1])/25) - 4320*E^((12*C[1])/25)*x + 216
0*E^((12*C[1])/25)*x^2)*#1^2 + (5832 + 8640*E^((12*C[1])/25) - 5832*x - 25920*E^
((12*C[1])/25)*x + 25920*E^((12*C[1])/25)*x^2 - 8640*E^((12*C[1])/25)*x^3)*#1^3 
+ (17496 + 19440*E^((12*C[1])/25) - 34992*x - 77760*E^((12*C[1])/25)*x + 17496*x
^2 + 116640*E^((12*C[1])/25)*x^2 - 77760*E^((12*C[1])/25)*x^3 + 19440*E^((12*C[1
])/25)*x^4)*#1^4 + (23328 + 23328*E^((12*C[1])/25) - 69984*x - 116640*E^((12*C[1
])/25)*x + 69984*x^2 + 233280*E^((12*C[1])/25)*x^2 - 23328*x^3 - 233280*E^((12*C
[1])/25)*x^3 + 116640*E^((12*C[1])/25)*x^4 - 23328*E^((12*C[1])/25)*x^5)*#1^5 + 
(11664 + 11664*E^((12*C[1])/25) - 46656*x - 69984*E^((12*C[1])/25)*x + 69984*x^2
 + 174960*E^((12*C[1])/25)*x^2 - 46656*x^3 - 233280*E^((12*C[1])/25)*x^3 + 11664
*x^4 + 174960*E^((12*C[1])/25)*x^4 - 69984*E^((12*C[1])/25)*x^5 + 11664*E^((12*C
[1])/25)*x^6)*#1^6 & , 1]^(-1)}, {y[x] -> 6 - 4*x + Root[16*E^((12*C[1])/25) + (
288*E^((12*C[1])/25) - 288*E^((12*C[1])/25)*x)*#1 + (729 + 2160*E^((12*C[1])/25)
 - 4320*E^((12*C[1])/25)*x + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (5832 + 8640*E^((
12*C[1])/25) - 5832*x - 25920*E^((12*C[1])/25)*x + 25920*E^((12*C[1])/25)*x^2 - 
8640*E^((12*C[1])/25)*x^3)*#1^3 + (17496 + 19440*E^((12*C[1])/25) - 34992*x - 77
760*E^((12*C[1])/25)*x + 17496*x^2 + 116640*E^((12*C[1])/25)*x^2 - 77760*E^((12*
C[1])/25)*x^3 + 19440*E^((12*C[1])/25)*x^4)*#1^4 + (23328 + 23328*E^((12*C[1])/2
5) - 69984*x - 116640*E^((12*C[1])/25)*x + 69984*x^2 + 233280*E^((12*C[1])/25)*x
^2 - 23328*x^3 - 233280*E^((12*C[1])/25)*x^3 + 116640*E^((12*C[1])/25)*x^4 - 233
28*E^((12*C[1])/25)*x^5)*#1^5 + (11664 + 11664*E^((12*C[1])/25) - 46656*x - 6998
4*E^((12*C[1])/25)*x + 69984*x^2 + 174960*E^((12*C[1])/25)*x^2 - 46656*x^3 - 233
280*E^((12*C[1])/25)*x^3 + 11664*x^4 + 174960*E^((12*C[1])/25)*x^4 - 69984*E^((1
2*C[1])/25)*x^5 + 11664*E^((12*C[1])/25)*x^6)*#1^6 & , 2]^(-1)}, {y[x] -> 6 - 4*
x + Root[16*E^((12*C[1])/25) + (288*E^((12*C[1])/25) - 288*E^((12*C[1])/25)*x)*#
1 + (729 + 2160*E^((12*C[1])/25) - 4320*E^((12*C[1])/25)*x + 2160*E^((12*C[1])/2
5)*x^2)*#1^2 + (5832 + 8640*E^((12*C[1])/25) - 5832*x - 25920*E^((12*C[1])/25)*x
 + 25920*E^((12*C[1])/25)*x^2 - 8640*E^((12*C[1])/25)*x^3)*#1^3 + (17496 + 19440
*E^((12*C[1])/25) - 34992*x - 77760*E^((12*C[1])/25)*x + 17496*x^2 + 116640*E^((
12*C[1])/25)*x^2 - 77760*E^((12*C[1])/25)*x^3 + 19440*E^((12*C[1])/25)*x^4)*#1^4
 + (23328 + 23328*E^((12*C[1])/25) - 69984*x - 116640*E^((12*C[1])/25)*x + 69984
*x^2 + 233280*E^((12*C[1])/25)*x^2 - 23328*x^3 - 233280*E^((12*C[1])/25)*x^3 + 1
16640*E^((12*C[1])/25)*x^4 - 23328*E^((12*C[1])/25)*x^5)*#1^5 + (11664 + 11664*E
^((12*C[1])/25) - 46656*x - 69984*E^((12*C[1])/25)*x + 69984*x^2 + 174960*E^((12
*C[1])/25)*x^2 - 46656*x^3 - 233280*E^((12*C[1])/25)*x^3 + 11664*x^4 + 174960*E^
((12*C[1])/25)*x^4 - 69984*E^((12*C[1])/25)*x^5 + 11664*E^((12*C[1])/25)*x^6)*#1
^6 & , 3]^(-1)}, {y[x] -> 6 - 4*x + Root[16*E^((12*C[1])/25) + (288*E^((12*C[1])
/25) - 288*E^((12*C[1])/25)*x)*#1 + (729 + 2160*E^((12*C[1])/25) - 4320*E^((12*C
[1])/25)*x + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (5832 + 8640*E^((12*C[1])/25) - 5
832*x - 25920*E^((12*C[1])/25)*x + 25920*E^((12*C[1])/25)*x^2 - 8640*E^((12*C[1]
)/25)*x^3)*#1^3 + (17496 + 19440*E^((12*C[1])/25) - 34992*x - 77760*E^((12*C[1])
/25)*x + 17496*x^2 + 116640*E^((12*C[1])/25)*x^2 - 77760*E^((12*C[1])/25)*x^3 + 
19440*E^((12*C[1])/25)*x^4)*#1^4 + (23328 + 23328*E^((12*C[1])/25) - 69984*x - 1
16640*E^((12*C[1])/25)*x + 69984*x^2 + 233280*E^((12*C[1])/25)*x^2 - 23328*x^3 -
 233280*E^((12*C[1])/25)*x^3 + 116640*E^((12*C[1])/25)*x^4 - 23328*E^((12*C[1])/
25)*x^5)*#1^5 + (11664 + 11664*E^((12*C[1])/25) - 46656*x - 69984*E^((12*C[1])/2
5)*x + 69984*x^2 + 174960*E^((12*C[1])/25)*x^2 - 46656*x^3 - 233280*E^((12*C[1])
/25)*x^3 + 11664*x^4 + 174960*E^((12*C[1])/25)*x^4 - 69984*E^((12*C[1])/25)*x^5 
+ 11664*E^((12*C[1])/25)*x^6)*#1^6 & , 4]^(-1)}, {y[x] -> 6 - 4*x + Root[16*E^((
12*C[1])/25) + (288*E^((12*C[1])/25) - 288*E^((12*C[1])/25)*x)*#1 + (729 + 2160*
E^((12*C[1])/25) - 4320*E^((12*C[1])/25)*x + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (
5832 + 8640*E^((12*C[1])/25) - 5832*x - 25920*E^((12*C[1])/25)*x + 25920*E^((12*
C[1])/25)*x^2 - 8640*E^((12*C[1])/25)*x^3)*#1^3 + (17496 + 19440*E^((12*C[1])/25
) - 34992*x - 77760*E^((12*C[1])/25)*x + 17496*x^2 + 116640*E^((12*C[1])/25)*x^2
 - 77760*E^((12*C[1])/25)*x^3 + 19440*E^((12*C[1])/25)*x^4)*#1^4 + (23328 + 2332
8*E^((12*C[1])/25) - 69984*x - 116640*E^((12*C[1])/25)*x + 69984*x^2 + 233280*E^
((12*C[1])/25)*x^2 - 23328*x^3 - 233280*E^((12*C[1])/25)*x^3 + 116640*E^((12*C[1
])/25)*x^4 - 23328*E^((12*C[1])/25)*x^5)*#1^5 + (11664 + 11664*E^((12*C[1])/25) 
- 46656*x - 69984*E^((12*C[1])/25)*x + 69984*x^2 + 174960*E^((12*C[1])/25)*x^2 -
 46656*x^3 - 233280*E^((12*C[1])/25)*x^3 + 11664*x^4 + 174960*E^((12*C[1])/25)*x
^4 - 69984*E^((12*C[1])/25)*x^5 + 11664*E^((12*C[1])/25)*x^6)*#1^6 & , 5]^(-1)},
 {y[x] -> 6 - 4*x + Root[16*E^((12*C[1])/25) + (288*E^((12*C[1])/25) - 288*E^((1
2*C[1])/25)*x)*#1 + (729 + 2160*E^((12*C[1])/25) - 4320*E^((12*C[1])/25)*x + 216
0*E^((12*C[1])/25)*x^2)*#1^2 + (5832 + 8640*E^((12*C[1])/25) - 5832*x - 25920*E^
((12*C[1])/25)*x + 25920*E^((12*C[1])/25)*x^2 - 8640*E^((12*C[1])/25)*x^3)*#1^3 
+ (17496 + 19440*E^((12*C[1])/25) - 34992*x - 77760*E^((12*C[1])/25)*x + 17496*x
^2 + 116640*E^((12*C[1])/25)*x^2 - 77760*E^((12*C[1])/25)*x^3 + 19440*E^((12*C[1
])/25)*x^4)*#1^4 + (23328 + 23328*E^((12*C[1])/25) - 69984*x - 116640*E^((12*C[1
])/25)*x + 69984*x^2 + 233280*E^((12*C[1])/25)*x^2 - 23328*x^3 - 233280*E^((12*C
[1])/25)*x^3 + 116640*E^((12*C[1])/25)*x^4 - 23328*E^((12*C[1])/25)*x^5)*#1^5 + 
(11664 + 11664*E^((12*C[1])/25) - 46656*x - 69984*E^((12*C[1])/25)*x + 69984*x^2
 + 174960*E^((12*C[1])/25)*x^2 - 46656*x^3 - 233280*E^((12*C[1])/25)*x^3 + 11664
*x^4 + 174960*E^((12*C[1])/25)*x^4 - 69984*E^((12*C[1])/25)*x^5 + 11664*E^((12*C
[1])/25)*x^6)*#1^6 & , 6]^(-1)}}

Maple raw input

dsolve((6-4*x-y(x))*diff(y(x),x) = 2*x-y(x), y(x),'implicit')

Maple raw output

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