AMATH-250-1055-Final_exam[1]

# AMATH-250-1055-Final_exam[1] - [10 1 Solve the following...

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1. Solve the following initial-value problem: [10] dy dx = 3 x 2 + 4 x + 2 2( y - 1) , y (0) = - 1 , to get an explicit expression for y ( x ). 2. (a) Use the integrating-factor method to solve the initial-value problem: [15] xy + 2 y = 4 x 2 , y (1) = 2 . (b) Give a qualitatively correct sketch of the 1-parameter family of solution curves representing the general solution found in (a). Highlight the solution satisfying the initial conditions. 3. Assume that the electrical power P generated by a wind turbine depends upon the [15] following physical quantities: ρ = density of air (mass per unit volume) v = air speed A = area swept out by turbine blades. [Power is work done per unit time.] (a) Construct the dimensional matrix for the problem, deduce the number of dimen- sionless variables available, and Fnd them. (b) Hence deduce how P depends functionally on ρ, ν , and A . (c) If the wind speed doubles, deduce the factor by which P changes. 4. Consider the following initial-value problem:

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## This note was uploaded on 10/24/2010 for the course AMATH 250 taught by Professor Ducharme during the Fall '09 term at Waterloo.

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AMATH-250-1055-Final_exam[1] - [10 1 Solve the following...

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