Question

Solve the ODE `(D-1)^2 (D^2+1)^2y=0` 

 

 

Answer 1

`(D-1)^2 (D^2+1)^2y=0 `  

For complementary solution , 

𝒇(𝑫)=𝟎  

`(D-1)^2(D^2+1)^2=0`

∴ (𝑫−𝟏)^𝟐=𝟎                  ∴ (𝑫^𝟐+𝟏)^𝟐=𝟎 

𝑫−𝟏=𝟎 𝒇𝒐𝒓 𝒕𝒘𝒐 𝒕𝒊𝒎𝒆𝒔              ` (D^2+1)=0`𝒇𝒐𝒓 𝒕𝒘𝒐 𝒕𝒊𝒎𝒆𝒔

∴ D -1=0                        ∴ 𝑫^𝟐=−𝟏 

Roots are : `D = 1,1,+i,+i,-i,-i` 

∴ `y_c=(c_1+xc_2)e^x+[(c_3+xc_4)cos x(c_5+xc_6)sin x]`