4.10.8 \((-y(x)+5 x+1) y'(x)-5 y(x)+x+5=0\)

ODE
\[ (-y(x)+5 x+1) y'(x)-5 y(x)+x+5=0 \] 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 = 0.0742956 (sec), leaf count = 925

\[\left \{\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,1\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,2\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,3\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,4\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,5\right ]}+1\right \},\left \{y(x)\to 5 x-\frac {1}{\text {Root}\left [\left (186624 e^{\frac {12 c_1}{25}} x^6+186624 x^4\right ) \text {$\#$1}^6+\left (-186624 e^{\frac {12 c_1}{25}} x^5-186624 x^3\right ) \text {$\#$1}^5+\left (77760 e^{\frac {12 c_1}{25}} x^4+69984 x^2\right ) \text {$\#$1}^4+\left (-17280 e^{\frac {12 c_1}{25}} x^3-11664 x\right ) \text {$\#$1}^3+\left (2160 e^{\frac {12 c_1}{25}} x^2+729\right ) \text {$\#$1}^2-144 e^{\frac {12 c_1}{25}} x \text {$\#$1}+4 e^{\frac {12 c_1}{25}}\& ,6\right ]}+1\right \}\right \}\]

Maple
cpu = 0.022 (sec), leaf count = 41

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

DSolve[5 + x - 5*y[x] + (1 + 5*x - y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (729 +
 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#1^3 
+ (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((12*C
[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 1]^(-1)
}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (729
 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#1^
3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((12
*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 2]^(-
1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + (7
29 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)*#
1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^((
12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 3]^
(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 + 
(729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^3)
*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*E^
((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & , 4
]^(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#1 
+ (729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*x^
3)*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 186624*
E^((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 & ,
 5]^(-1)}, {y[x] -> 1 + 5*x - Root[4*E^((12*C[1])/25) - 144*E^((12*C[1])/25)*x*#
1 + (729 + 2160*E^((12*C[1])/25)*x^2)*#1^2 + (-11664*x - 17280*E^((12*C[1])/25)*
x^3)*#1^3 + (69984*x^2 + 77760*E^((12*C[1])/25)*x^4)*#1^4 + (-186624*x^3 - 18662
4*E^((12*C[1])/25)*x^5)*#1^5 + (186624*x^4 + 186624*E^((12*C[1])/25)*x^6)*#1^6 &
 , 6]^(-1)}}

Maple raw input

dsolve((1+5*x-y(x))*diff(y(x),x)+5+x-5*y(x) = 0, y(x),'implicit')

Maple raw output

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