4.10.31 $(x-3y(x))y^{\prime }(x)-y(x)+3x+4=0$

ODE
$$(x-3y(x))y^{\prime }(x)-y(x)+3x+4=0$$ ODE Classification

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

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

Mathematica ✓
cpu = 0.0688683 (sec), leaf count = 781

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

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

$$\{-\frac{2}{3}\ln \left(\frac{-2y\left(x\right)-4-2x}{3+2x}\right)-\frac{1}{3}\ln \left(\frac{-2y\left(x\right)+2+2x}{3+2x}\right)-\ln (3+2x)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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