Question

Consider the following statements. (i) 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. (ii) We must rethink what we mean by solving y"' + y' - y = { cos(x + 42) * # 1 before trying to compute a solution defined on an interval containing x = = 1. (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) The procedure of finding series solutions to a homogeneous linear second-order ODES could be accurately described as the “method of undetermined series coefficients”. (v) The particular solution of the ODE y'' – 4y' + 4y Cxe2x where C is an undetermined constant. (vi) 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. 6e2x is given by yp = Determine which of the above statements are True (1) or False (2).

Consider the following statements. (i) 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. (ii) We must rethink what we mean by solving y′′ + y′ − y  =  { cos(x + 42) x  ≠  1 0 x  =  1 before trying to compute a solution defined on an interval containing x  =  1. (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) The procedure of finding series solutions to a homogeneous linear second-order ODEs could be accurately described as the “method of undetermined series coefficients”. (v) The particular solution of the ODE y′′ − 4y′ + 4y  =  6e2x is given by yp  =  Cxe2x where Cis an undetermined constant. (vi) 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.

Answer 1

O A Taylore series gives the expansion of a function around any point of it is given as 2 fewa fca) + (x-9). f'ra) + (-a) f"9) + f(a) 3! TRUE statement. When value of x=1 then differential eq Changes from non- Homogeneous to Honogenesus and its solution will Comprise of only Complem. mentry function gives as D²D-1=0 7 D = 1 + Vite -1 +15 = 2 C.F. = 4.6% 1 * (5) 2 6 ) TRUE statement. CS Scatteredith CamScanner

1) No information gives about ketule. Bus the series solution of gives differential equation could be obtained with the help of method of undetermined series Coefficient by obtained fremulae TRUE securetence 22 ♡ The dift. egon gives as y"-4y+4y= be 0² 40+4=0 Ausiliasy equation Partiwlar solution (pos.) = 6622 D²_4D+4 D360+g=0 for a=2 P. Sa 2x 6. de for a=3=2. 2D - 4 Ps= 3 x ² e 2x 20-4=0 for D=3 Pallicellor solution obtained does not matches Р. with creer FALSE, which = 3 2² e ²x ♡ FALSĒTh Camscanner