Section 3.3 High-Order Nonhomogeneous Linear ODE

eqn=y'''[x]+3 y''[x]+3y'[x]+y[x]==30 Exp[-x]

DSolve[eqn,y[x],x]

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

sol=DSolve[{eqn,y[0]==3,y'[0]==-3,y''[0]==-47},y,x]

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

Plot[f[x],{x,0,10}]

eqn=x^3 y'''[x]-3x^2y''[x]+6x y'[x]-6y[x]==x^4 Log[x]

DSolve[eqn,y[x],x]

eqn=y''''[x]==f0/(Ey Iz)

sol=DSolve[{eqn,y[0]==0,y[L]==0,y''[0]==0,y''[L]==0},y[x],x]

Factor[%]

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

Ey=1;Iz=1;L=1;f0=1;

f[x]

Plot[-f[x],{x,0,1},PlotRange->{{0,1},{-0.1,0.1}}]