solve the following equation (d^4 y)/(dx^4 )+r^2 y=0

Rewriting the given equation in the symbolic form, we have:
(D^4+r^4 )y=0
So  A.E. is                                  m^4+r=0
Or                        (m^2+r^2+√2 mr)(m^2+r^2-√2 mr)=0
                (m^2+r^2+√2 mr)=0  and  (m^2+r^2-√2 mr)=0

Now         m^2+√2 mr+r^2=0   gives  m=-r/√2±r/√2 i
and           m^2-√2 mr+r^2=0   gives    m=r/√2±r/√2 i
Therefore,
〖Y_(G.S.)=Y〗_(C.F.)=
e^(-rx/√2) {c_1  cos⁡ (rx/√2)+c_2  sin⁡ (rx/√2)  }
+e^(rx/√2) {c_3  cos⁡ (rx/√2)+c_4  sin⁡ (rx/√2)  }
Where c_1, c_2, c_3, c_4 are arbitrary constants.