4.9.19 \(x \log (x) y'(x)=a x (\log (x)+1)-y(x)\)

ODE
\[ x \log (x) y'(x)=a x (\log (x)+1)-y(x) \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.00652861 (sec), leaf count = 16

\[\left \{\left \{y(x)\to a x+\frac {c_1}{\log (x)}\right \}\right \}\]

Maple
cpu = 0.006 (sec), leaf count = 14

\[ \left \{ y \relax (x ) =ax+{\frac {{\it \_C1}}{\ln \relax (x ) }} \right \} \] Mathematica raw input

DSolve[x*Log[x]*y'[x] == a*x*(1 + Log[x]) - y[x],y[x],x]

Mathematica raw output

{{y[x] -> a*x + C[1]/Log[x]}}

Maple raw input

dsolve(diff(y(x),x)*x*ln(x) = a*x*(1+ln(x))-y(x), y(x),'implicit')

Maple raw output

y(x) = a*x+1/ln(x)*_C1