Answered: "4th order linear ordinary differential equation?"

Ayden Case

Answered question

2022-03-01

4th order linear ordinary differential equation?

chezpepina87j

Beginner2022-03-02Added 6 answers

Step 1

λ^{4} - λ^{2} + 1

can be completed to a square by variation of the middle term, making it more negative in the process

= {(λ^{2} + 1)}^{2} - 3 λ^{2} = (λ^{2} + \sqrt{3} λ + 1) (λ^{2} - \sqrt{3} λ + 1)

Now one can apply the usual solution formulas for quadratic equations (with real coefficients but complex roots).

par5o4nr4z

Beginner2022-03-03Added 5 answers

Step 1
The characteristic equation of you ODE is

x^{4} - x^{2} + 1 = 0

which, when multiplied in

x^{2} + 1

, yields

x^{6} + 1 = 0

.
Hence, the roots of

x^{4} - x^{2} + 1

are those of

x^{6} = - 1

except those of

x^{2} = - 1

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