Question

Solve `(D^3+1)^2y=0`

Answer 1

`(D^3+1)^2y=0`

For complementary solution,
f(D) = 0
`therefore(D^3+1)^2=0`

`therefore(D^3+1)=0`

Roots are : `"D"=-1,1/2+isqrt3/2,1/2-isqrt3/2`  .. for two times

Roots of given diff. eqn are real and complex.
The general solution of given diff. eqn is given by ,

`y_g=y_c=(c_1+xc_2)e^(-x)+e^(x/2)[(c^3+xc_4)cosx+(c_5+xc_6)sinx`