##### 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 = 0.335558 (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 = 0.367 (sec), leaf count = 98

$\left [y \left (x \right ) = \RootOf \left (b \,\textit {\_Z}^{2}-{\mathrm e}^{\RootOf \left (x^{2} \left (\left (\tanh ^{2}\left (\frac {\sqrt {a^{2}-4 b}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 a}\right )\right ) a^{2}-4 \left (\tanh ^{2}\left (\frac {\sqrt {a^{2}-4 b}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 a}\right )\right ) b -4 b \,{\mathrm e}^{\textit {\_Z}}-a^{2}+4 b \right )\right )}+1+a \textit {\_Z} \right ) x\right ]$ 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))

Maple raw output

[y(x) = RootOf(b*_Z^2-exp(RootOf(x^2*(tanh(1/2*(a^2-4*b)^(1/2)*(2*_C1+_Z+2*ln(x)
)/a)^2*a^2-4*tanh(1/2*(a^2-4*b)^(1/2)*(2*_C1+_Z+2*ln(x))/a)^2*b-4*b*exp(_Z)-a^2+
4*b)))+1+a*_Z)*x]