Question

Evaluate `(d^4y)/(dx^4)+2(d^2y)/(dx^2)+y=0`

Answer 1

put `d/dx=D`

`therefore D^4y+2D^2y+y=0`

`therefore D^4+2D^2+1=0`

Put `D^2=t`

`=> t^2+2t+1=0`

`=> t=-1, -1`

Roots are : D=+i,-i,+i,-i
The complementary solution of given eqn is

`y_c=y_g=(C_1+xC_2)cosx+(C_3+xC_4)sinx`