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lecture_18 (dragged) 1 - 2 MA 36600 LECTURE NOTES: FRIDAY,...

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2 MA 36600 LECTURE NOTES: FRIDAY, FEBRUARY 27 Summary. We summarize the general solution to the diferential equation ay °° + by ° + cy =0 . Say that the characteristic equation ar 2 + br + c =0 has roots r 1 and r 2 . A Fundamental set oF solutions For this constant coefficient diferential equation is { y 1 ,y 2 } in terms oF the Functions y 1 ( t )= e r 1 t iF b 2 4 ac> 0, e r 1 t iF b 2 4 ac = 0, e λt cos μt iF b 2 4 ac< 0. y 2 ( t )= e r 2 t iF b 2 4 ac> 0, te r 1 t iF b 2 4 ac = 0, e λt sin μt iF b 2 4 ac< 0. Perhaps this result does not look symmetric when either b 2 4 ac is positive or negative. We make some observations to remedy this. Recall that Euler’s ±ormulas imply e it = cos t + i sin t e it = cos t i sin t ° = cos t = e it + e it 2 , sin t = e it e it 2 i . Hence we can express the trigonometric Functions in terms oF complex-valued exponential Functions. Simi- larly, we de²ne the hyberbolic
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