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Solve the Ode ( D − 1 ) 2 ( D 2 + 1 ) 2 Y = 0 - Applied Mathematics 2

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Solve the ODE `(D-1)^2 (D^2+1)^2y=0` 

 

 

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Solution

`(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                  ∴ (𝑫𝟐+𝟏)𝟐=𝟎 

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

∴ D -1=0                        ∴ 𝑫𝟐=−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]` 

 

Concept: Linear Differential Equation with Constant Coefficient‐ Complementary Function
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