ODE classification

Nasser M. Abbasi

May 7, 2017 compiled on — Sunday May 07, 2017 at 12:35 AM
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1 examples

1.1 second order, constant coeff.

1.1.1 second order, constant coeff. homogeneous

second order, constant coeff. homogeneous, one root repeated

y′′ − 2y ′ + 1 = 0

Let y = Aerx  and plug into the above and simplify, we obtain the charaterstic equation

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Repated root. Hence the two L.I. basis solutions are

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And the homogeneous solution is

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second order, constant coeff. homogeneous, two real distinct roots

 ′′    ′
y + y  − 2y = 0

Let y = Aerx  and plug into the above and simplify, we obtain the charaterstic equation

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Hence the L.I. basis solutions are

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And the homogeneous solution is

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second order, constant coeff. homogeneous, two complex conjugate roots

 ′′     ′
y − 6y  + 13y = 0

Let        rx
y = Ae  and plug into the above and simplify, we obtain the charaterstic equation

r2 − 6r + 13 = 0

Whose roots are

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Hence the L.I. basis solutions are

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the homogeneous solution is

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This can be converted to real basis using Euler relation which results in

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1.1.2