TITLE: Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$
QUESTION [8 upvotes]: When I was solving some differential equations, I asked myself the following:
Is there a function has the following:
$$y'=y+1$$
$$y''=y+1$$
$$y'''=y+1$$
$$......$$
$$......$$
If the initial value is $$y(0)=1$$

REPLY [6 votes]: All your differential equations after the first one are implied by the first equation. If $y'=y+1$ then differentiation of it gives $y''=y'$ and further differentiation gives the successive equations. So it is really just about solving the first equation with the given initial value, and this you can do.