Milestones_1_Math_33B_F08

Milestones_1_Math_33B_F08 - Mathematics 33B: Milestones for...

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Mathematics 33B: Milestones for Week 1 Book sections: 2.1, 2.2, 2.4, 2.5. TOPIC 1: Basic definitions Read the first chapter and know what is meant by the normal form of a differential equation, an initial value problem, and the interval of existence. Example: The equation x 00 = - g is in normal form. Example: ± x 00 = - g x 0 (0) = v 0 , x (0) = x 0 Is an initial value problem and its solution is x ( t ) = x 0 + v 0 t - g 2 t 2 . The value of x (0) and x 0 (0) are called initial values, they are several values given at one point. Boundary values for example might be the values x (0) and x ( . 42), values given at two or more points. A problem where a differential equation is given with boundary values is called a boundary value problem. Example: Interval of existence. The book example is nice, we use it. Consider the initial value problem y 0 = y 2 with y (0) = 1. A solution is - 1 t - 1 . The interval of existence is ( -∞ , 1). TOPIC 2: Direction Fields. You should know how to geometrically interpret the solutions of a differ- ential equation. We look at the following examples in class: Example: y 0 = 1. Solution: y ( t ) = t + C. Example: The exponential equation,
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Milestones_1_Math_33B_F08 - Mathematics 33B: Milestones for...

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