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# home1_09F - NOT use a computer code to make the sketch...

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Department of Mathematical Sciences University of Delaware Prof. T. Angell September 9, 2009 Mathematics 351 Exercise Sheet 1 Exercise 1 : For each of the following di ff erential equations, determine values of r for which the di ff erential equation has solutions of the form y = e rt . (a) y + 2 y = 0 , (b) y + y 6 y = 0 . Exercise 2 : For each of the following di ff erential equations, determine values of r for which the di ff erential equation has solutions of the form y = t r . (a) t 2 y + 4 ty + 2 y = 0 , (b) t 2 y 4 ty + 2 y = 0 . Exercise 3 : Show that the function ϕ ( x ) 1 is a solution of the di ff erential equation y = 1 y 3 and sketch a direction field for this equation on a sheet of graph paper. Do
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Unformatted text preview: NOT use a computer code to make the sketch. Exercise 4 : Show that the equation y 2 ty = 1 has solution y ( t ) = e t 2 t e s 2 ds + e t 2 . Exercise 5 : (a) Show that the one parameter family of functions ( x ; K ) := Ke 2 x is a family of solutions of the equation y = 2 y . (b) Find a constant A so that ( x ) A satises y + 2 y = 6. Check that, for this value of A , the two parameter family A + Ke 2 x also is a set of solutions of y + 2 y = 6. Due Date: Wednesday, Sept. 16, in class...
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