Hello, I have two questions about this problem:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] (D^4 + 5D^2 + 4)y = 0 [/tex]

y(0) = 10

y'(0) = 10

y''(0) = 6

y'''(0) = 8

[tex] \lambda^4 - 5\lambda^2 + 4 = 0 [/tex]

[tex] (\lambda^2 + 4) (\lambda^2 + 1) [/tex]

Until here I am fairly sure that I didn't mess it up..

But I'm not sure if I have the roots correct. Are the roots:

Root1:

[tex] \lambda^2 = -4 [/tex]

[tex] \lambda = \pm\sqrt-4 [/tex]

[tex] \lambda = \pm\sqrt-1 \cdot \pm\sqrt4[/tex]

[tex] \lambda = \pm2i [/tex]

Root2:

[tex] \lambda^2 = -1 [/tex]

[tex] \lambda = \pm\sqrt-1 [/tex]

[tex] \lambda = \pm1i [/tex]

So this is a complex double root (I think?) and the equation should be:

[tex] y = e^{\gamma x} [ (A_1 + A_2x) cos\omegax + (B_1 + B_2x) sin\omegax ] [/tex]

Making it:

[tex] y = e^{(0)x} [ (A_1 + A_2x) cos2x + (B_1 + B_2x) sinx ] [/tex]

* [Question 1]: Is this the right way to proceed?

Because, I finished the problem and it turned out to be wrong... (and it took a long time to differentiate it 3 times too.. and even longer to check it again.. twice.. *sigh*)

Also, I'm not sure exactly how I am supposed to find [tex] \inline \gamma [/tex] in [tex] \inline e^{\gammax} [/tex]...

Is it right to think that if the root is [tex] \inline \lambda = 5 \pm 3i [/tex] then [tex] \inline \gamma [/tex] is 5 and [tex] \inline \omega [/tex] is 3? (making the equation:)

[tex] y = e^{5x} [ (A_1 + A_2x) cos3x + (B_1 + B_2x) sin\omegax ] [/tex]

But this must be wrong because what if the other root has a value added to the multiple of the i value too?

* [Question 2] How do you find [tex] \inline \gamma [/tex] ?

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# Homework Help: Higher Order ODE - Multiple Complex Roots?

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