4.10.31 \((x-3 y(x)) y'(x)-y(x)+3 x+4=0\)

ODE
\[ (x-3 y(x)) y'(x)-y(x)+3 x+4=0 \] 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 = 0.0688683 (sec), leaf count = 781

\[\left \{\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,1\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,2\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,3\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,4\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,5\right ]}\right )\right \},\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (16 e^{12 c_1}+1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664\right )+\text {$\#$1}^4 \left (-384 x^4-2304 x^3-5184 x^2-5184 x-1944\right )+\text {$\#$1}^3 \left (64 x^3+288 x^2+432 x+216\right )+\text {$\#$1}^2 \left (36 x^2+108 x+81\right )+\text {$\#$1} (-12 x-18)+1\& ,6\right ]}\right )\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 55

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

DSolve[4 + 3*x - y[x] + (x - 3*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 43
2*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*
#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9
216*x^5 + 1024*x^6)*#1^6 & , 1]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1
 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 -
 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*
x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 2]^(-1))/3
}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 
432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4
)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 +
 9216*x^5 + 1024*x^6)*#1^6 & , 3]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*
#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944
 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 4665
6*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 4]^(-1))
/3}, {y[x] -> (x - Root[1 + (-18 - 12*x)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 
+ 432*x + 288*x^2 + 64*x^3)*#1^3 + (-1944 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x
^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46656*x + 77760*x^2 + 69120*x^3 + 34560*x^4
 + 9216*x^5 + 1024*x^6)*#1^6 & , 5]^(-1))/3}, {y[x] -> (x - Root[1 + (-18 - 12*x
)*#1 + (81 + 108*x + 36*x^2)*#1^2 + (216 + 432*x + 288*x^2 + 64*x^3)*#1^3 + (-19
44 - 5184*x - 5184*x^2 - 2304*x^3 - 384*x^4)*#1^4 + (11664 + 16*E^(12*C[1]) + 46
656*x + 77760*x^2 + 69120*x^3 + 34560*x^4 + 9216*x^5 + 1024*x^6)*#1^6 & , 6]^(-1
))/3}}

Maple raw input

dsolve((x-3*y(x))*diff(y(x),x)+4+3*x-y(x) = 0, y(x),'implicit')

Maple raw output

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