F21 Math 2045 Exam 5 Prep Worksheet

Math 2045 Exam 5 Prep Worksheet Chapter 11. 1, 11.2, 10.5, 11.4 Topic Examples: 1.Classify a differential equation.  Separable DE: the DE can be written as q ( y ) dy = p ( x ) dx  Linear first order DE: the DE can be written as dy dx + P ( x ) y = Q ( x ) Linear Separable a) dy dx = 3 x + 1 y b) dy dx = y + x c) y ' = e x ( y − 1 ) d) y ' = √ y e) y ' = x 2 − y 2 2.Write DE for application problems a) Exponential growth/decay model: the population is growing/decreasing at a rate proportional to the population dy dt = k ∙ y b) Limited growth model: the population is growing at a rate proportional to the difference between the maximum N and the population dy dt = k ( N − y ) Logistic growth model: the population is growing at a rate proportional to the product of the population and the fractional difference between the maximum N and the population dy dt = k ∙ y ( 1 − y N ) Example: a) A new cell phone is introduced into the market. It is predicted that sales will grow logistically. The manufacturer estimates that they can sell a maximum of 70 thousand cell phones. After 29 thousand cell phones have been sold, sales are increasing by 4 thousand phones per month. Find the differential equation describing the cell phone sales, where y = y ( t ) is the number of cell phones (in thousands) sold after t months. b) The amount of a tracer dye injected into the bloodstream decreases exponentially, with a decay constant 3% per minute. Write the differential equation describing the situation, where y is the amount of the tracer dye after t minutes. c) An isolated fish population is limited to 4000 by the amount of food available. If there are now 320 fish and the population is growing with a growth constant of 2% a year, write the differential equation to describe the situation, where y is the fish population after t years. 3.Given solution to a differential equation, find the unknown parameter Find the values of k for which the function y ( x ) = e kx − 2 is a solution of the differential equation y ' ' − 8 = 4 y

4.Solve differential equations by separation of variables. Find the general solution dy dx = y 2 5 √ x 5.Solve linear first-order differential equations by using the integrating factor. The amount a full-time student is educated ( x ) changes with respect to the student's age t according to the differential equation dx dt = 1 − kx , where k is a constant. Suppose x ( 0 ) = x 0 . 6. Perform matrix operations 2 ae − 3 t [ 1 − 5 ] + bt [ − 2 6 ] 7.Find eigenvalues and eigenvectors for a 2x2 matrix [ − 255 1152 − 56 253 ]

8.Write systems of linear differential equations for compartment problems in matrix form 9.Solve a system of linear differential equations using eigenvalues and eigenvectors