4.11.5 \(y'(x) (a x+b y(x))=a y(x)+b x\)

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

[[_homogeneous, `class A`], _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 = 0.0368521 (sec), leaf count = 45

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

Maple
cpu = 0.016 (sec), leaf count = 50

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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