Math 2410Q - 10 Elementary Differential Equations Summer 2017 Midterm Exam Review Guide
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1 Math 410Q - 10 Elementary Differential Equations Summer 017 Mierm Exam Review Guide Math 410Q Mierm Exam Info: Covers Sections questions in total Some questions will have multiple parts. 1 of the 7 questions will be a set of True/False questions, where you will need to justify your answer if it s false. 1 of the 7 questions will be a Bonus Question Duration will be 60 minutes Possible topics on the Mierm Exam: Sec. Topic 1.1 -Identify the independent and dependent variable(s in a differential equation. -Identify the order of a differential equation -Classify a differential equation as ordinary or partial, and autonomous or non-autonomous. 1. -Find the general solution to a separable differential equation. -Find the solution to an initial value problem with a separable differential equation. -Solve a mixing problem: Set up an initial value problem modeling the rate of change of the amount of a substance in a container, and determine the amount of substance in the container at time t. 1.3 Describe briefly the behavior of the solution that satisfies y(t 0 = y 0 as t or as t, when given the slope field of a differential equation. 1.4 Use Euler s method with a given step size to approximate the solution of a first-order initial value problem over an interval. (The answer should include a table of approximate values and a sketch of the solution. 1.5 Show understanding of the Existence Theorem and the Uniqueness Theorem of Solutions Find the equilibrium points of an autonomous differential equation. -Sketch the phase line of an autonomous differential equation. -Classify each equilibrium point as a sink, source, or node. -Determine the long-term behavior of the solution that satisfies an initial condition y(t 0 = y 0. (See the last example in Section 1.6 notes Find the bifurcation values of a given one-parameter family of differential equations. -Draw the bifurcation diagram containing phase lines (in numerical order for values of the parameter smaller than, larger than, and equal to the bifurcation values Classify a differential equation as linear homogeneous, linear nonhomogeneous, or nonlinear. -Show understanding of the Linearity Principle for Linear Differential Equations (for both homogeneous and nonhomogeneous differential equations Find the general solution of a first order linear differential equation using integrating factors. -Solve an initial value problem using integrating factors. -Use Newton s Law of Cooling to find the temperature of an object at a certain time t. -Set up and solve a first-order linear differential equation to find the amount of money in a savings account after t years..1 Classify first-order systems of differential equations as linear, linear homogeneous, linear nonhomogeneous, nonlinear, and/or autonomous.. Sketch the direction field of a given system of differential equations at several given points..6 Show understanding of the Existence and Uniqueness Theorem for Systems. (Turn over for page.
2 Sec. Topic 3.1 -Determine whether a given linear homogeneous system has a nontrivial equilibrium solution or only the trivial equilibrium solution. -Show understanding of the Linearity Principle for Systems (Principle of Superposition. -Determine if given vectors are linearly independent or linearly dependent. -Show that two vectors Y 1 (t and Y (t are solutions to a given system d Y = A Y. -If Y 1 (t and Y (t are linearly independent at t 0, find constants c 1 and c such that the solution Y (t = c 1Y1 (t + c Y (t satisfies Y (t 0 = (x 0, y Compute the eigenvalues of the coefficient matrix of a system, and compute an eigenvector for each eigenvalue. -Find the general solution to a system d Y = A Y, where A has distinct, real, nonzero eigenvalues. -Find the solution Y (t that satisfies Y (t 0 = (x 0, y Classify the origin (the trivial solution as a sink, source, or saddle using the eigenvalues of the coefficient matrix. Tips: Carefully stu all definitions and theorems. *This is important for True/False questions, and to justify your answers.* Stu and review the examples in the lecture notes. Stu and redo the homework problems. Stu the in-class quizzes and their solutions. Stu the problems on the Mierm Exam Review (see next page.
3 Math 410Q - 10 Elementary Differential Equations Summer 017 Mierm Exam Review Problems 1. For the following differential equations, (i State the dependent variable and the independent variable(s. (ii Classify it as an ordinary differential equation or a partial differential equation. (iii State the order. If the differential equation is ordinary and of order 1, determine if it is linear homogeneous, linear nonhomogeneous, or nonlinear. (a 1 + t y + sin(t d y = 5e t (b e t y + 4y + 6y = cos(t (c Let y = y(x, w be a function of two variables. 1 + cos(x + y ( y x + 1 (d = cos(ty + t + 1 (e x y + xy + (x c y = e x, where c is a constant. (f Let f = f(x, y be a function of two variables. (g + ln t y = 0 y x = 3 y w 3 f x + f y + xy = x + y. Find the general solution to the following first-order differential equations. (a = et y (b = ty 1+y (c t = y ln(y (d y 3 dx = x + dx (e + ( t y = cos(t t (f y t+1y = (t Find the solution to the following initial-value problems. (a = t(1+y y, y(0 = 1 (b = sin(t y, y ( π = 1 (c = t 1+y, y( 1 = 0 (d = y + et, y(0 = 3 (e dx + 3x x +1 y = 6x x +1, y(0 = 1 (f t y = cos(t 1+t, y ( π = 0
4 4. A tank contains 100 gallons of pure water. At time zero, a sugar-water solution containing 0. lb of sugar per gallon enter the tank at a rate of 3 gallons per minute. Simultaneously, a drain is opened at the bottom of the tank allowing the sugar solution to leave the tank at 3 gallons per minute. Assume that the solution in the tank is kept perfectly mixed at all times. (a What will be the sugar content in the tank after 0 minutes? (b Eventually, what will be the sugar content in the tank after a very long time? 5. The Jones family is saving money for tuition so that their young daughter, Jessica, can attend a private high school in Manhattan. Suppose that Jessica s parents have annual savings of $1, 000 a year that is deposited into a savings account. Assume that the account earns interest at a rate of 3% annually compounded continuously, and that her parents begins with $3, 000 in their savings account. Assuming that the family does not spend any money from this account, how much money will they have saved after 10 years? 6. Dr. Alvarez is sitting in her office where the thermostat reads 0 C. She decides to pour herself a (non-insulated cup of tea from a kettle whose contents are 60 C. Suppose the heating constant of her cup of tea is (a Let y(t denote the temperature of her cup of tea at time t, where t is in minutes. Using Newton s Law of Cooling, set up an initial-value problem modeling the rate of change of the temperature of the tea. (b Find the temperature of her tea after 15 minutes. Round to decimal places. 7. Suppose a cup of tea has a temperature of 180 F in a kitchen whose temperature is 7 F. (a Let y(t denote the temperature of the tea at time t, where t is in minutes. Use Newton s Law of Cooling to set up an initial value problem in terms of k that models the temperature of the tea. (b After two minutes, the temperature of the tea is 150 F. Find the value of the cooling constant k. Round to decimal places. (c Find the temperature of the tea after 5 minutes. 8. Use Euler s method with step size t = 0.5 to approximate the solution to the initial-value problem = t ty, y(0 = 8 over the interval 0 t. Your answer should include a table of the approximate values of the dependent variable and include a sketch of the approximate solution. Round each y k value to 1 decimal place. k t k y k (t k, y k 9. Consider the differential equation = y (y 1(y + 4. (a Draw the phase line for the given differential equation. (b Classify each equilibrium point as a sink, source, or node. (c Describe the long-term behavior of the solution that satisfies the initial condition y(0 = π.
5 10. Consider the following one-parameter family = y 4y + k with parameter k. (a Identify the bifurcation values. (b Draw the bifurcation diagram containing phase lines for the values of the parameter smaller than, larger than, and equal to the bifurcation values. Be sure to arrange the phase lines in numerical order in terms of k. 11. Consider the following linear system of differential equations: dy ( 5 4 = A Y, where A = 8 7 (a Compute the eigenvalues of A. (b Find an eigenvector for each eigenvalue found in part (a. (c Find the general solution of the system. (You do not need to verify linear independence. (d Classify the origin as a sink, source, or saddle. 1. Consider the following linear system of differential equations: dy ( 6 8 = A Y, where A = 0 (a Compute the eigenvalues of A. (b Find an eigenvector for each eigenvalue found in part (a. (c Find the general solution of the system. (You do not need to verify linear independence. (d Classify the origin as a sink, source, or saddle. 3
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