4.10.45 \((6 y(x)-x+5) y'(x)=4 y(x)-x+3\)

ODE
\[ (6 y(x)-x+5) y'(x)=4 y(x)-x+3 \] 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 = 0.18332 (sec), leaf count = 1177

\[\left \{\left \{y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}+1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}-\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}-1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x+\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}+1}-5\right )\right \},\left \{y(x)\to \frac {1}{6} \left (x-\frac {2 (x+1)}{\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}} x+\sqrt {-\frac {3 \cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+3 \sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+2}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right )}+\sqrt {-\frac {\cosh \left (\frac {4 c_1}{9}\right )+\sinh \left (\frac {4 c_1}{9}\right )}{(x+1)^2 \left (\cosh \left (\frac {4 c_1}{9}\right ) (x+1)^2+\sinh \left (\frac {4 c_1}{9}\right ) (x+1)^2+1\right ){}^2}}+\frac {3}{(x+1)^2}}-1}-5\right )\right \}\right \}\]

Maple
cpu = 0.03 (sec), leaf count = 43

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

DSolve[(5 - x + 6*y[x])*y'[x] == 3 - x + 4*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (-5 + x + (2*(1 + x))/(1 + Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9]
 + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh
[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1
])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))
] + x*Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2
*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 
+ x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^
2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))]))/6}, {y[x] -> (-5 + x - (2*(
1 + x))/(-1 + Sqrt[3/(1 + x)^2 - Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((
1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2
 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + 
(1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))] + x*Sqrt[3/(1 + x)^2 
- Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4
*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9]
 + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1
 + x)^2*Sinh[(4*C[1])/9]))]))/6}, {y[x] -> (-5 + x + (2*(1 + x))/(1 + Sqrt[3/(1 
+ x)^2 + Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*
Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C
[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/
9] + (1 + x)^2*Sinh[(4*C[1])/9]))] + x*Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(4*C[1])/
9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Si
nh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C
[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]
))]))/6}, {y[x] -> (-5 + x - (2*(1 + x))/(-1 + Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(
4*C[1])/9] + Sinh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 +
 x)^2*Sinh[(4*C[1])/9])^2))] - (2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*S
inh[(4*C[1])/9])/((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*
C[1])/9]))] + x*Sqrt[3/(1 + x)^2 + Sqrt[-((Cosh[(4*C[1])/9] + Sinh[(4*C[1])/9])/
((1 + x)^2*(1 + (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9])^2))] - 
(2 + 3*(1 + x)^2*Cosh[(4*C[1])/9] + 3*(1 + x)^2*Sinh[(4*C[1])/9])/((1 + x)^2*(1 
+ (1 + x)^2*Cosh[(4*C[1])/9] + (1 + x)^2*Sinh[(4*C[1])/9]))]))/6}}

Maple raw input

dsolve((5-x+6*y(x))*diff(y(x),x) = 3-x+4*y(x), y(x),'implicit')

Maple raw output

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