please, I need solutions on a word document file and all questions answered.
Microsoft Word - Final Exam- MATH 282- Differential Equation Instructor: Dr. Giro Carbone Final Exam MATH 282 Differential Equations, Spring 2021 Show all your work, Total number points = 80 you need to get 72 or above for an A. MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 1 1) A large tank is filled to capacity with 300 gallons of pure water. Brine containing 4 pounds of salt per gallon is pumped into the tank at a rate of 3 gal/min. The well-mixed solution is pumped out at the same rate. Find the number A(t) of pounds of salt in the tank at time t. (10 points) MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 2 2) a. Find the general solution of the given differential equation. (8 points) 22dy y x x dx x b. Give the largest interval over which the general solution is defined. (Express your answer using interval notation.) (1 points) c. Identify any transient terms in the general solution, if there are none, state NONE.) (1 points) MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 3 3) Find two power series solutions of the given differential equation about the ordinary point . Compute the first three terms for each series (10 points). 0y xy MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 4 4) Initially 10 grams of a radioactive substance was present. It has a half-life of 52 hours. The rate of decay is proportional to the amount present. a) Give the IVP that models this problem. Let A be the amount present, ? be the time, and ? be the constant of proportionality. (2 points) b) Solve the DE that you found in part a). Your answer will be a function in terms of ? and ?. (4 points) c) How much of the substance is left after 10 hours? (4 points) MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 5 5) The following MATLAB code performs Euler’s method for an IVP. Note: MATLAB starts indexing with 1 and not 0! When we do Euler’s method by hand, we call our initial values ? and ? . MATLAB calls its initial values ? and ? . So, for example, if by hand we calculated (? , ? ), (? , ? ), …, (? ,? ) using Euler’s method, MATLAB would calculate the same six points but they would be labeled (? , ? ), (? , ? ), …, (? ,? ). Questions on the next page. MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 6 5) Problem continued (refer to the MATLAB code above) a) What IVP is being used in this code? (4 points) b) What is the final value for x (ie. what is ? )? (4 points) c) Suppose you want to change only the IVP being used in this code example. What lines would you need to change? (Just give the line numbers.) (2 points) MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 7 6) Use the Laplace Transform to solve the given initial-value problem. (10 points) 3 ( 1), (0) 0y y t yu MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 8 7) A mass of 1 slug is attached to a spring whose constant is 5 lb/ft. Initially, the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s, and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity. (Take the downward direction to be positive.) a) Using the method of undetermined coefficients, find the equation of motion if the mass is driven by an external force equal to ( ) 12cos(2 ) 3sin(2 )f t t t b) What is the steady-state solution for this system? MATH 282 Spring 2021, Dr. G. Carbone, Student’s Name__________________ 9 8) The number ?(?) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem. (10 points) (1 0.0002 ), (0) 1 dN N N N dt a) Use the phase portrait concept to predict how many supermarkets are expected to adopt the new procedure over a long period of time. b) Solve the initial-value problem given above. c) How many supermarkets are expected to adopt the new technology when ? = 15?