4.11.2 \(y'(x) (a x+b y(x))+x=0\)

ODE
\[ y'(x) (a x+b y(x))+x=0 \] ODE Classification

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

Book solution method
Change of Variable, new independent variable

Mathematica
cpu = 0.377994 (sec), leaf count = 73

\[\text {Solve}\left [\frac {a \tan ^{-1}\left (\frac {a+\frac {2 b y(x)}{x}}{\sqrt {4 b-a^2}}\right )}{\sqrt {4 b-a^2}}+\frac {1}{2} \log \left (\frac {a x y(x)+b y(x)^2+x^2}{x^2}\right )+\log (x)=c_1,y(x)\right ]\]

Maple
cpu = 0.017 (sec), leaf count = 67

\[ \left \{ -{\frac {1}{2}\ln \left ({\frac {axy \relax (x ) +b \left (y \relax (x ) \right ) ^{2}+{x}^{2}}{{x}^{2}}} \right ) }+{a{\it Artanh} \left ({\frac {ax+2\,by \relax (x ) }{x}{\frac {1}{\sqrt {{a}^{2}-4\,b}}}} \right ) {\frac {1}{\sqrt {{a}^{2}-4\,b}}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x + (a*x + b*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[(a*ArcTan[(a + (2*b*y[x])/x)/Sqrt[-a^2 + 4*b]])/Sqrt[-a^2 + 4*b] + Log[x] 
+ Log[(x^2 + a*x*y[x] + b*y[x]^2)/x^2]/2 == C[1], y[x]]

Maple raw input

dsolve((a*x+b*y(x))*diff(y(x),x)+x = 0, y(x),'implicit')

Maple raw output

-1/2*ln((a*x*y(x)+b*y(x)^2+x^2)/x^2)+a/(a^2-4*b)^(1/2)*arctanh((a*x+2*b*y(x))/x/
(a^2-4*b)^(1/2))-ln(x)-_C1 = 0