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 = 0.221946 (sec), leaf count = 781
\[\left \{\left \{y(x)\to \frac {1}{3} \left (x-\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 (1024 x^6+9216 x^5+34560 x^4+69120 x^3+77760 x^2+46656 x+11664+16 e^{12 c_1}\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 = 0.249 (sec), leaf count = 242
\[\left [y \left (x \right ) = -\frac {1}{2}-\frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-144 \left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}-\frac {3456 \left (3+2 x \right )^{3} \textit {\_C1}}{\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+864 i \sqrt {3}\, \left (\frac {\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {4 \left (3+2 x \right )^{3} \textit {\_C1}}{\left (108 \left (3+2 x \right )^{3} \textit {\_C1} +12 \sqrt {-96 \left (3+2 x \right )^{9} \textit {\_C1}^{3}+81 \left (3+2 x \right )^{6} \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{2985984 \textit {\_C1}}-\left (3+2 x \right )^{3}}{2 \left (3+2 x \right )^{2}}\right ]\] 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 + 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 & , 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]) + 46656*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 + (-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 & , 6]^(-1))/3}}
Maple raw input
dsolve((x-3*y(x))*diff(y(x),x)+4+3*x-y(x) = 0, y(x))
Maple raw output
[y(x) = -1/2-1/2*(1/2985984/_C1*(-1/2+1/2*I*3^(1/2))^6*(-144*(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)-3456*(3+2*x)^3*_C1/(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)+864*I*3^(1/2)*(1/6*(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)-4*(3+2*x)^3*_C1/(108*(3+2*x)^3*_C1+12*(-96*(3+2*x)^9*_C1^3+81*(3+2*x)^6*_C1^2)^(1/2))^(1/3)))^2-(3+2*x)^3)/(3+2*x)^2]