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midterm_2_review (dragged) 1 - 2 MA 36600 MIDTERM #2 REVIEW...

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Unformatted text preview: 2 MA 36600 MIDTERM #2 REVIEW • Say that y1 = y1 (t) and y2 = y2 (t) are solutions to the homogeneous equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = 0. The Principle of Superposition states that the linear combination y (t) = c1 y1 (t) + c2 y2 (t) is also a solution for any constants c1 and c2 . We say that {y1 , y2 } is a Fundamental Set of Solutions for the differential equation if this is the general solution. • Let y1 = y1 (t) and y2 = y2 (t) be any two functions. They are said to be linearly independent if the equation “c1 y1 (t) + c2 y2 (t) = 0 for all t” has the unique solution c1 = c2 = 0. Otherwise, we say y1 and y2 are linearly dependent. • Let y1 = y1 (t) and y2 = y2 (t) be any two differentiable functions. The Wronskian of y1 and y2 is ￿ ￿ ￿ ￿ y (t) y2 (t) ￿ ￿ W y1 , y2 (t) = y1 (t) y2 (t) − y1 (t) y2 (t) = det 1 . ￿ ￿ y1 (t) y2 (t) • Say that y1 = y1 (t) and y2 = y2 (t) are solutions to the homogeneous equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = 0. Abel’s Theorem states that their Wronskian is ￿￿ ￿ ￿ W y1 , y2 (t) = C exp − t b(τ ) dτ a(τ ) ￿ for some constant C . This implies that the following statements are equivalent: i. y1 (t) and y2 (t) are linearly independent. ii. {y1 , y2 } form a fundamental set of solutions to the homogeneous equation. ￿ ￿ iii. W ￿y1 , y2 ￿(t0 ) ￿= 0 for some t0 . iv. W y1 , y2 (t) ￿= 0 for all t. If any of these statements are true, then y (t) = c1 y1 (t) + c2 y2 (t) is the general solution. §3.3: Complex Roots of the Characteristic Equation. • Euler’s Formula is the expression ert = eλt cos µt + i eλt sin µt where r = λ + i µ. • Consider the constant coefficient differential equation a y ￿￿ + b y ￿ + c y = 0. Say that the quadratic polynomial a r2 + b r + c has discriminant b2 − 4 a c < 0 i.e., the characteristic equation has complex roots ￿ r1 = λ + i µ |b2 − 4 a c| b in terms of λ=− , µ= . r2 = λ − i µ 2a 2a Then real-valued functions y1 (t) = eλt cos µt and y2 (t) = eλt sin µt form a fundamental set of solutions. • The trigonometric functions may be expressed in terms of complex-valued exponentials: ￿ eit = cos t + i sin t eit + e−it eit − e−it =⇒ cos t = , sin t = . 2 2i e−it = cos t − i sin t Similarly, the hyberbolic functions may be expressed in terms of real-valued exponential functions: ￿t e = cosh t + sinh t et + e−t et − e−t cosh t = , sinh t = =⇒ −t 2 2 e = cosh t − sinh t In general, when given the constant coefficient equation a y ￿￿ + b y ￿ + c y = 0, define the quantities ￿ |b2 − 4 a c| b λ=− and µ= . 2a 2a ...
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