4.4.3 \(x y'(x)=x^n \log (x)-y(x)\)

ODE
\[ x y'(x)=x^n \log (x)-y(x) \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.0206706 (sec), leaf count = 29

\[\left \{\left \{y(x)\to \frac {c_1}{x}+\frac {x^n ((n+1) \log (x)-1)}{(n+1)^2}\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 31

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

DSolve[x*y'[x] == x^n*Log[x] - y[x],y[x],x]

Mathematica raw output

{{y[x] -> C[1]/x + (x^n*(-1 + (1 + n)*Log[x]))/(1 + n)^2}}

Maple raw input

dsolve(x*diff(y(x),x) = x^n*ln(x)-y(x), y(x),'implicit')

Maple raw output

y(x) = 1/(n+1)*ln(x)*x^n-x^n/(n+1)^2+1/x*_C1