Study Guide for Chapters 1-3
Study Guide for Chapters 1-3 MAT 275
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This 5 page Study Guide was uploaded by Darren Smith on Monday September 19, 2016. The Study Guide belongs to MAT 275 at Arizona State University taught by Lopez in Fall 2016. Since its upload, it has received 136 views.
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Date Created: 09/19/16
Darren Smith Not For Resale Chapters 1.1-3.4 Review 1.1 Direction Fields Be able to Draw/Recognize slope fields When drawing you can: Set y’ equal to some constant and solve for where x is = to that slope To Recognize slopes: Set y = to 0 to see what the slope is like along x-axis and the same for x. General form for population growth: dp/dt = rp-k where r is the growth rate, k is the predation rate, and p(t) is the population function. Be able to solve for constant solutions of a differential equation and for the values are increasing or decreasing Be able to identify if a differential equation is stable, semi-stable, or unstable Stable: Graph of a function crosses the x-axis from positive to negative values Semi-Stable: Graph of a function touches the x-axis but doesn’t pass through Unstable: Graph of a function goes through x-axis from negative to positive values Ex1) Draw a direction field for the equation y’=t + 2y (Pg. 10) Ex2) What are the constant solutions of dy/dt = –y^4 + 3y^3 + 28y^2 and for what values of y is y increasing? (WebWork 1.1 Problem 5) 1.3 Classification of Differential Equations Be able to determine whether an equation is linear or not and what order it is Order=the highest power derivative Linear?=Check if the derived variable has a power of 2 or more Be able to match differential equations with their solutions Ex1) Verify that y=3t + t^2 is a solution to the differential equation ty’ – y = t^2 (pg. 25) 2.1 Integrating Factor Works with first order equations Write the equation in the standard form: dy/dt + p(t)y = g(t) Find the integrating factor, u(t) = e^ integral(p(t)) Darren Smith Not For Resale Plug into the equation 1/u(t) * integral(u(t)g(t)dt + c/u(t) Ex1) y’ + 3y = t + e^(-2t) find the general solution to this problem, and use it to find how it behaves as t infinity. (pg. 40) 2.2 Separable Equations First order equations Write in the standard form: M(x,y) + N(x,y)dy/dx = 0 Multiply by dx to separate variables in order to integrate. Solve for y Know how to solve these using initial conditions Ex) Solve dy/dx = (x-e^x)/(y+e^y) (pg. 48) 2.3 Modeling With First Order Equations Be able to work with word problems to formulate a differential equation for some sort of rate For water of a certain concentration flowing in and out at the same time: dQ/dt = rate in – rate out where t is time and Q is the amount of salt concentration For investment problems: dS/dt = rS where r is the interest rate and S(t) is the current value of the investment If you want to know the equation for a body of constant mass affected by gravity look at page 59 Ex) A tank originally contains 100 gal of fresh water. Then water containing ½ lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and frewsh water is poured into the tank at a rate of 2 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (pg. 60) Darren Smith Not For Resale 2.5 Population Dynamics Be able to tell when the function is increasing/decreasing Be able to tell what will happen as tinfinity Given an initial population, know what will happen before it reaches equilibrium Verhulst’s equation: dP/dt = kP(1-P/K) where K is the carrying capacity/saturation level, k is the intrinsic growth rate, and P(t) is the current population Gompertz equation: dP/dt = c*ln(K/P)P where c is a constant and K is the carrying capacity For examples look at pg. 90 of the textbook. 2.7 Euler Method y n+1 = yn+ f(tn,n )h where h is the step size and f() is found by plugging in the initial values to the function given. Ex) Find approximate values for the solution of the given initial value problem y’ = 3+t-y where h=.1 and y(0) = 1. (pg.110) 2.8 Existence and Uniqueness Address the following questions: Do all the functions in the sequence [phi(k)] exist? (this is out of the notes) Does the sequence converge? Ratio Test What are the properties of the limit function? Does it satisfy the differential equation? Is this the only solution? Take the abs value of the difference of phi(t) and some solution equation. If the solution plugged into phi(t) returns phi(t), it is also a solution. Ex) Do all that for y’ = 2t(1+y), y(0) = 0 (check notes on 2.8 for the example) Darren Smith Not For Resale 3.1 Homogeneous Constant Coefficients nd 2 Order linear equations General Form: Y’’ + p(t)y’ + q(t)y = g(t) Must be equal to zero: Y’’ + p(t)y’ + q(t)y = 0 If we plug in e^(rt) to the second equation we get: 2 ar + br + c = 0 which is the characteristic equation Should be solvable to the following form: Y = C 1 1t) + 2 2 (t) Use initial conditions to solve for the Cs Ex) Find the solution of y’’ + y’ – 2y = 0 when y(0) = 1, y’(0) = 1 (pg. 144) 3.3 Complex Roots 2 2 This happens when b – 4ac of the equation ar + br + c = 0 is negative Solve for the roots, and check that their wronskian doesn’t equal 0 Know the proof that e^(it) = cos(t) + isin(t) For ex: y(t) = exp[(-1/2 + 3i)te =t(cos(3t) + isin(3t)) =1y Y2would just be the same equation with (cos(3t) – isin(3t)) The general solution to these equations is: Y(t) = 1 1 (t) +2 2u (t) For the example equation, let u 1t) = ½(y 1t) + y2(t)) = e^(-1/2*t)*cos(3t) and 2 (t) = 1/(2i)*(y1(t) – 2 (t)) = e^(-1/2*t)*sin(3t) Then substitute back into the general solution equation with the u’s. lambda*t Ultimately, the important equation is y(t) = e (c1cos(ut) + c2sin(ut) Ex) Find the general solution to y’’ + 2y’ – 8y = 0 (pg.164) Darren Smith Not For Resale 3.4 Repeated Roots This is when b^2 = 4ac, meaning r =r =-b1(22) (from notes) The equation needed is y(t) = e (c + c t1 2 Ex) Find the general solution of y’’-2y’+10y = 0 (pg. 172) *Know the principle of Superposition (3.2) *Know how to find the Wronskian determinant(3.2) These may or may not be on the test but they would be good to know
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