4.11.26 \(x (y(x)+4) y'(x)=y(x)^2+2 y(x)+2 x\)

ODE
\[ x (y(x)+4) y'(x)=y(x)^2+2 y(x)+2 x \] ODE Classification

[_rational, [_Abel, `2nd type``class B`]]

Book solution method
Homogeneous equation, special

Mathematica
cpu = 0.0220779 (sec), leaf count = 96

\[\left \{\left \{y(x)\to \frac {1}{\frac {1}{x+4}-\frac {\left (\frac {x}{x+4}\right )^{3/2}}{x \sqrt {\frac {c_1 (x+4)-4}{x+4}}}}-4\right \},\left \{y(x)\to \frac {1}{\frac {\left (\frac {x}{x+4}\right )^{3/2}}{x \sqrt {\frac {c_1 (x+4)-4}{x+4}}}+\frac {1}{x+4}}-4\right \}\right \}\]

Maple
cpu = 0.033 (sec), leaf count = 74

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

DSolve[x*(4 + y[x])*y'[x] == 2*x + 2*y[x] + y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -4 + ((4 + x)^(-1) - (x/(4 + x))^(3/2)/(x*Sqrt[(-4 + (4 + x)*C[1])/(4 
+ x)]))^(-1)}, {y[x] -> -4 + ((4 + x)^(-1) + (x/(4 + x))^(3/2)/(x*Sqrt[(-4 + (4 
+ x)*C[1])/(4 + x)]))^(-1)}}

Maple raw input

dsolve(x*(4+y(x))*diff(y(x),x) = 2*x+2*y(x)+y(x)^2, y(x),'implicit')

Maple raw output

1/(4+y(x))+x^(1/2)/(4+x)^(3/2)/(_C1-4/(4+x))^(1/2)-1/(4+x) = 0, 1/(4+y(x))-x^(1/
2)/(4+x)^(3/2)/(_C1-4/(4+x))^(1/2)-1/(4+x) = 0