For more explanation about the difference between the following equations.

Question

Can someone tell me the difference between the two following initial value problems :

$$ \begin{array}{l} D^\alpha x(t)=f(t,x(t)) \\ x(0)=x_0 \end{array} \tag 1 $$

$$ \begin{array}{l} D^\alpha (x(t)-x_0)=f(t,x(t)) \\ x(0)=x_0 \end{array} \tag 2 $$

Thanks a lot in advance

Answer 1

There is no difference, because $$D^{\alpha}(x(t)-x_0)=D^{\alpha}x(t)-D^{\alpha}x_0=D^{\alpha}x(t)$$