Section 3.1 High-Order Homogeneous Linear ODE

eqn=y''''[x]- 5y''[x]+4 y[x]==0

DSolve[eqn,y[x],x]

eqn=x^3 y'''[x]-3x^2 y''[x]+6x y'[x]-6 y[x]==0

sol=DSolve[eqn,y[x],x]

y''[1]=.

sol=DSolve[{eqn,y[1]==2,y'[1]==1,y''[1]==-4},y,x]

s[x_]:=c1 x + c2 x^2 +c3 x^3

eqn1=s[1]==2

eqn2=s'[1]==1

eqn3=s''[1]==-4

Solve[{eqn1,eqn2,eqn3},{c1,c2,c3}]

eqn=y''''[x]-6y''[x]+25y[x]==0

Solve[λ^4-6λ^2+25==0,λ]

DSolve[eqn,y[x],x]