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
\[ \left \{ -3\,\ln \left ( {\frac {3-x-y \left ( x \right ) }{-1+x}} \right ) +2\,\ln \left ( {\frac {-y \left ( x \right ) +4-2\,x}{-1+x}} \right ) -\ln \left ( -1+x \right ) -{\it \_C1}=0 \right \} \] 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^((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 + 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 - 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 & , 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])/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 & , 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) - 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 & , 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 + 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 & , 5]^(-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 + 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