Section 3.2 High-Order Homogeneous ODE with Constant Coefficients

eqn=y'''[x]- y''[x]+100 y'[x]-100 y[x]==0

DSolve[eqn,y[x],x]

y'[0]=.;y''[0]=.;

sol=DSolve[{eqn,y[0]==4,y'[0]==11,y''[0]==-299},y,x]

f[x_]:=y[x] /. sol

Plot[f[x],{x,0,3}]

eqn=y'''''[x]-3y''''[x]+3y'''[x]-y''[x]==0

DSolve[eqn,y[x],x]

eqn=y''''[x]-29y''[x]+100y[x]==0

DSolve[eqn,y[x],x]