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Unformatted text preview: Homework assignment 1 * Due date: September 18 1. Determine the values of n for which the following equation is exact, and solve the equation for those values: ( x + y e 2 xy ) dx + nx e 2 xy dy = 0 2. Bernoullis equation Bernoulli s equation: dy dx + p ( x ) y = q ( x ) y n is linear if (and only if) n equals 0 or 1. Nevertheless, show that the transformation z = y 1 n yields a linear equation. Use this transformation to solve: y + xy = xy 4 3. The catenary (aka the shape of a hanging cable) Imagine a power cable whose ends are attached to 2 poles of the same height. The poles are stuck in the ground, which we assume forms a nice flat and horizontal surface. The weight of the cable is uniform and we denote the weight of the cable per unit length by w . The horizontal tension in the midpoint of the cable (where the cable is horizontal) is denoted by T . Introduce a Cartesian coordinate system ( x, y ) with origin on the ground right below the midpoint of the cable. The coordinate of this midpoint is denoted by (0 , y ). We want to find the height y ( x ) of each point of the cable. It can be shown (dont do this!) that y satisfies the following differential equation:...
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- Summer '06