MATH 251 Work sheet Things to know Chapter 1 1 Classifications of differential equation ODEs vs PDEs What is the order of an equation What makes an equation linear Ex 1 1 1 Which equation is linear What is each equation s order a y 5t3 y ln t y t e 2t b y 2t y y4 c y 3 y 4 sin 2t y 0 d y tan t2 y f t Ex 1 1 2 Can you give an example of a fourth order linear equation A sixth order nonlinear equation 2011 Zachary S Tseng 1 2 Direction fields What is it How to draw it Ex 1 2 1 y t2 y What is an integral curve Trace the integral curve that passes through the point 1 1 2011 Zachary S Tseng 2 MATH 251 Work sheet Things to know Chapter 2 1 First order linear equations Standard form Ex 2 1 1 y 9y 6 Some as the above example could be solve by re arrangement but there is a general method that of integrating factor that solves this type of equations Know the formulas of the integrating factor method When using them first be sure that the equation is in its standard form Integrating factor t General solution y t 2011 Zachary S Tseng 3 Ex 2 1 2 t3 y 4t2y 5e 2t Standard form is t y t Ex 2 1 3 see Ex 1 2 1 y t2 y Standard form is t y t Ex 2 1 4 Use the integrating factor method to solve the equation in example 2 1 1 2011 Zachary S Tseng 4 2 Initial Value Problems I V P What are the 2 components of an IVP Its solution s if exists contains no arbitrary constant coefficient particular solution Know the difference between a general solution and a particular solution Ex 2 2 1 Take the previous example t3 y 4t2y 5e 2t Find its solution satisfying i y 1 1 ii y 1 1 The Existence and Uniqueness Theorem for first order linear equations What is it How to find the largest interval also known as the interval of validity on which a particular solution is guaranteed to exist uniquely Continue with the example above find the largest interval on which the particular solution is guaranteed to exist uniquely for each of parts i and ii 2011 Zachary S Tseng 5 3 Separable equations The key to solving one is to rewrite the equation into the form F y dy G x dx Then integrate both sides Simplify if necessary but be wary of division by zero The form above is a way to identify the type too If a first order equation can be rewritten into this form then it is a separable equation Otherwise it is not Ex 2 3 1 Ex 2 1 1 The linear equation y 9y 6 is also separable Why Implicit solution vs explicit solution How do they differ Ex 2 3 2 y x2y2 x2 y 3 1 How to separate Implicit solution is Explicit solution is Particular solution is explicitly 2011 Zachary S Tseng 6 Ex 2 3 3 Identify whether each equation below is separable Solve the equation if it is separable a y xy 5 b y xy 4x y 4 2011 Zachary S Tseng 7 4 Applications of first order equations I Solution mixing Necessary information parameters are rate in rate out concentration in initial quantity of solute and initial volume The volume is given by S t The generic mixing equation is It is always a linear equation sometimes it is separable as well Ex 2 4 1 Exam I spring 2003 Certain bacteria cell contains 2 nl cytoplasm This bacterium is placed in a substance polluted with 3ng nl of certain harmful chemical The bacterium exchanges fluids with its environment at a rate of 1 100 nl s That is there is a flow of 1 100 nl s of the polluted substance into the bacterium and the well mixed solution flows out of the bacterium at the same rate Assume that the cytoplasm is initially free of the chemical a Find a formula for the amount of the chemical in the cell at any given time t 0 b The bacterium is going to die when the concentration of the chemical reaches 3 ng nl When is this going to happen 2011 Zachary S Tseng 8 Note that the equation in the previous example is both linear and separable It can be solved as either type using the appropriate method Ex 2 4 2 A mixing tank is initially filled with 300 liters of pure methanol Additional methanol containing 10 grams per liter of iodine flows into the mixing tank at a rate of 5 liters per minute The well stirred iodine solution flows out of the vat at a rate of 6 liters per minute Find the amount of dissolved iodine in the mixing tank at any time t until the tank is completely drained of the solution What is the maximum amount of iodine present in the tank during this process 2011 Zachary S Tseng 9 II Motion with air resistance Necessary information parameters mass drag coefficient initial velocity Simple linear equation model of an object undergoing free fall Equation of velocity How to find its displacement function Limiting velocity What is it How to find it A nonlinear equation model of motion Equation It is a separable equation solve it accordingly 2011 Zachary S Tseng 10 III Continuous compound interest Necessary information parameters interest growth rate net transaction per unit time principal initial amount Equation It is a linear equation Make sure the rate and net transaction have the same unit of time Ex 2 4 3 A college student has incurred 50000 in student loans Upon graduation t 0 the loans are to be repaid in equal monthly payments over 15 years Assume the interest rate is fixed 5 per year compounded continuously What will his monthly payment be 2011 Zachary S Tseng 11 5 Autonomous equations and stability of equilibrium solutions Autonomous equations What are they Every autonomous equation is always separable Therefore we know how to solve them Equilibrium solutions How to find them Stability classification What are the 3 types of stability What are their computational significances Know how to visualize stability using direction fields How to determine the stability of an equilibrium solution There are several methods Perhaps the easiest one to remember is a modified version of the first derivative test from calculus 2011 Zachary S Tseng 12 Ex 2 5 1 Find and classify all equilibrium solutions of y y 1 y 4 2 y 9 3 Ex 2 5 2 Find and classify all equilibrium solutions of y 1 sin 2y 2011 Zachary S Tseng 13 6 Exact equations Standard form with fine print Be sure to review partial differentiation and integration before proceeding Verification how to tell whether an equation is exact Ex 2 6 1 Is the equation 2x 2y 2x 2y y 0 an exact equation Why or why not How to solve an exact …
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