# ordinary differential equations – Solutions to the ODE \$exp(D)x=0\$

Let $$D$$ denote the differential operator and let $$exp(D)$$ denote the exponential function of the differential operator, that is, $$exp(D)=I+D+D^2/2+…$$. What are the solutions to the ODE $$exp(D)x=0$$? Put differently, what are the solutions to the ODE $$x+x’+x”/2+…=0$$