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Unformatted text preview: ) = ° t f ( s, φ n ( s )) ds. Homogeneous equations with constant coeﬃcients ²irst we will review the basic concepts linear/nonlinear equations, homogeneous/nonhomogeneous equations. ²ollow these steps to approach an initial value problem ay °° + by ° + cy = 0 , y ( t ) = y , y ° ( t ) = y ° . Step 1: Solve the characteristic equation ar 2 + br + c = 0. Assume that it has two diFerent real roots r 1 and r 2 . (More complicated situations will be discussed in later sections.) Step 2: The solution will be y = c 1 e r 1 t + c 2 e r 2 t , where c 1 = y ° − y r 2 r 1 − r 2 e − r 1 t c 2 = y r 1 − y ° r 1 − r 2 e − r 2 t 1 Examples 5. φ ( t ) = ° ∞ k =2 2 − k +2 k ! ( − t ) k . Variation of parameters, # 38 on page 41. 25. (a) y = 1 5 (1 + 2 β ) e − 2 t + 1 5 (4 − 2 β ) e t/ 2 . (b) t m = 2 ln(6) / 5. (c) β = 2. 2...
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 Spring '08
 Fonken
 Approximation, #, Picard, integral equation, 2 2 K, r2 y0 r1

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