SCHOOL OF MATHEMATICAL SCIENCES ENG1091 Mathematics for Engineering Assignment 6: Solutions of ODEs and Innite Series Due date: Lab class for week 11 There are two questions in this assignment worth a total of 20 marks (5% of the nal mark for this unit). The rst question (10 marks) is an exam style question. The second question (10 marks) is a traditional assignment style question. The questions will be marked using distinct criteria as detailed in the unit-guide and in the examples on the unit website. Please read these carefully. For full marks you will need to show all of your working. Late submissions will be subject to late penalties (see the unit guide for full details). Short answer exam style question 1. (a) Use l'Hopitals rule to evaluate the limit lim x1 5/10 1 x + loge x 1 + cos(x) (b) Calculate the same limit as in part (a) but this time use suitable Taylor series for the numerator and denominator. 5/10 School of Mathematical Sciences Monash University Detailed answer assignment style question 2. Remember how we played with innite series some lectures ago? In that game we computed the coecients in the series expansion of a function like sin(x) by computing successive derivatives. Suppose now that that function is a solution of an ODE. Could that ODE be used to help compute those coecients? Indeed it can. This opens up a new way to solve dierential equations - commonly known as a series solution of the dierential equation. You will nd a vast array of material on this method in the library and on the internet (it is not covered in the lecture notes nor in Glynn James). In this question you will construct the series solution of the following ODE dy = 2y dx Of course we are making a big assumption here - that the solution y(x) can be written as a Taylor series. But let's not let that stop us having some fun. (a) Assume that y(0) = A for some given number A. Use the ODE to compute the successive derivatives of y(x) at x = 0. 2/10 (b) Use the result of part (a) to deduce the coecients a0 , a1 , a2 , a3 and an in the innite series 2/10 y(x) = a0 + a1 x + a2 x2 + a3 x3 + + an xn + (c) Use the result of part (b) to write down y(x) as an innite series. Your answer should contain only one constant A and should include the general term in the innite series. 3/10 (d) Solve the ODE using any other method. Does this solution agree with what you found in part (c)? Justify your answer. 3/10 16-Feb-2015 2