MEC702_Wk_10_Lect_1

# MEC702_Wk_10_Lect_1 - MEC702 Wk 10 Lect 1 Background on...

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Unformatted text preview: MEC702 Wk 10 Lect 1 Background on Partial Fractions There is a theorem in advanced algebra which states that every proper rational fraction can be expressed as a sum P (x) = F1 (x) + F2 (x) + . . . + Fn (x) Q(x) where F1 (x), F2 (x), . . . , Fn (x) are rational functions of the form A (ax + b)k or Ax + B + bx + c)k (ax2 in which the denominators are factors of Q(x). This sum is called the fraction decomposition. Procedure to decompose a partial fraction 1. Factorize the denominator Q(x) partial P (x) Q(x) completely, into linear and irreducible quadratic factors. 2. For each of the linear factors occurring once (ax + b) once, it is written as A . ax + b 3. For the linear factors occurring twice (ax + b)2 , then it is written as B A + ax + b (ax + b)2 and this goes on with occurring thrice and so on. 4. For the irreducible quadratic factor occuring once written as as Ax + B . ax2 + bx + c 1 (ax2 + bx + c), it is 5. Once all the factors have been listed down, the capitalized variables are solved. Example. Decompose the following into partial fractions. 1. 1 x2 +x−2 2. 2x+4 x3 −2x2 3. x2 +x−2 3x3 −x2 +3x−1 Example. Solve the IVP y 00 + y 0 − 2y = 0, Solution. y(0) = 4, y 0 (0) = −5 . y(0) = 3, y 0 (0) = −3.5. y(t) = et + 3e−2t . Example. Solve the IVP y 00 + y 0 + 0.25y = 0, Solution. y(t) = y = 3e−0.5t − 2te−0.5t . Example. Solve the IVP y 00 − y = t, Solution. y(0) = 1, y 0 (0) = 1. y(t) = et + sinh t − t. Example. Solve the IVP y 00 + y 0 + 9y = 0, Solution. y(0) = 0.16, y 0 (0) = 0 . y(t) = e0.5t (0.16 cos 2.96t + 0.027 sin 2.96t). 2 ...
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