Question

Problem #2: Consider the following statements. [6 marks) (1) The particular solution of the ODE)" - 6y' + 9y = 5e3x is given by yp = Cre3x where C is an undetermined constant. (ii) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the "method of undetermined series coefficients". (iii) Most of the material in Lecture Notes from Week 3 to Week 5, inclusive, can be extended or generalized to higher-order ODES (iv) A Taylor series is a power series that gives the expansion of a function around a point a. Convergence of such series is fully understood by means of the ratio test. (v) The underlying idea behind the method of undetermined coefficients is a conjecture about the form of a particular solution that is motivated by the right-hand side of the equation. The method of undetermined coefficients is limited to second-order linear ODEs with constant coefficients and the right-hand side of the ODE cannot be an arbitrary function. (vi) We must rethink what we mean by solving S cos(x +44) X1 y"+y'-y = X = 1 y = { before trying to compute a solution defined on an interval containing x = 1. Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True False False,True False, True then you would enter '1,2,2,1,2,1' into the answer box below (without the quotes),

Answer 1

os, 34 Se her a=3 .P.S= 3x 5xe i Porticulas solution oblained (6) Differentiel equelon frien y"- 60+ 3y = sex susiliary equation: B-ost9 = 0 Particular solution (P.S) = D² - 6D+ 9 for { xex matches with 2D-6 B-6D+ 9 = 0 Q=3=D P.S. 3x x e 20-6=0 2. for D=3 which does not 2 32 cae FALSE stotiment

recurrence care no A cond 21 31 The series solution solution of given differential equation could be obtained with the help method undetermined series Coefficients" by obtaining formule "TRUE statement , Since Lecture notes not available in the given problem Comment . Taylor serius ques the expansion of a functions around cony point a it is quen as, f(a) + (x-0) f'a) + (x-g²f" (a) + (x-e) f" (a) + : TRUE statement , The method of undetermined cool coefficients is an approach to finding a particular solution to certain oulinary differential electione e Mucurrince relations coefficiente should be constant and side right they can be veriable of differential equation could be any cerbitrary function FALSI statement (vi) when value of Then differential equation changes from homogenous to homependes to its solution will comprese of only complementary function given as, D² +D-1=0 * 1.1174 -1 + 5 +1+sa с. е statement non en hangenen So it is not necessary that cie of x=1 non D = -lt 2 2 (+1-Sla t, e. C.F. = as TRUE