4.5.26 \((x+1) y'(x)=n y(x)+e^x (x+1)^{n+1}\)

ODE
\[ (x+1) y'(x)=n y(x)+e^x (x+1)^{n+1} \] ODE Classification

[_linear]

Book solution method
Linear ODE

Mathematica
cpu = 0.012048 (sec), leaf count = 17

\[\left \{\left \{y(x)\to \left (c_1+e^x\right ) (x+1)^n\right \}\right \}\]

Maple
cpu = 0.007 (sec), leaf count = 14

\[ \left \{ y \relax (x ) = \left ({{\rm e}^{x}}+{\it \_C1} \right ) \left (1+x \right ) ^{n} \right \} \] Mathematica raw input

DSolve[(1 + x)*y'[x] == E^x*(1 + x)^(1 + n) + n*y[x],y[x],x]

Mathematica raw output

{{y[x] -> (1 + x)^n*(E^x + C[1])}}

Maple raw input

dsolve((1+x)*diff(y(x),x) = exp(x)*(1+x)^(n+1)+n*y(x), y(x),'implicit')

Maple raw output

y(x) = (exp(x)+_C1)*(1+x)^n