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

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.0298682 (sec), leaf count = 83

\[\left \{\left \{y(x)\to -\frac {\sqrt {a^2 x^2-b^2 x^2+b e^{2 c_1}}+a x}{b}\right \},\left \{y(x)\to \frac {\sqrt {a^2 x^2-b^2 x^2+b e^{2 c_1}}-a x}{b}\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 36

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

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

Mathematica raw output

{{y[x] -> -((a*x + Sqrt[b*E^(2*C[1]) + a^2*x^2 - b^2*x^2])/b)}, {y[x] -> (-(a*x)
 + Sqrt[b*E^(2*C[1]) + a^2*x^2 - b^2*x^2])/b}}

Maple raw input

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

Maple raw output

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