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2.8 &amp; 3.1

# 2.8 &amp; 3.1 - = ° t f s φ n s ds Homogeneous...

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Sections 2.8 & 3.1 Existence and Uniqueness This section introduces Picard’s method of successive approximations. You don’t need to understand every detail of the discussion following Example 1, especially item 1. Uniform convergence is the central idea behind items 2 & 3. The technique introduced in item 4 (also shown in Example 1) is commonly used in calculus, but is not the main emphasis of this course. Things that do merit attention in this section are: equivalence of di ff erential equation and integral equation: y = f ( t, y ) , y (0) = 0 φ ( t ) is a solution to the above equation as long as it satisfies φ ( t ) = t 0 f ( s, φ ( s )) ds. Notice that the initial value restriction is embedded in the integral equation, as we set the lower limit of integration to be 0. There are many variations of this correspondence, but again, they are not the main emphasis of this course. So don’t delve too much into it. Picard’s iteration formula:

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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, homoge-neous/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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2.8 &amp; 3.1 - = ° t f s φ n s ds Homogeneous...

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