Question

Please answer #4

Problem 3: Find a solution to the IVP dy dy + dc2 + y = 0, y(0) = y'(0) = 1. dx Problem 4: Suppose you are given the differential equation ay" +by' + cy = 9(2) where a, b, and c are constants. For each of the following choices of g(x), write down the form for the particular solution Yp that you would use: (a) g(x) = 205 (b) g(x) = x²e32 (c) g(x) = xº cos(x) (d) g(x) = cos(x) + cos(2x) + cos(3.c)

Answer 1

Ans: 4) ay" +by' + cy = g(x), where ai 6 and C are constants. a) 91x):45 The Pasticulas + Anxh solution form - Ao+ Aix + whese Ao, A, . An are constant b) g(x)= x² 3x particulas solution form = 3* ( Aot A1 x + A₂ x², whese Ao, A, and Az are constants. The C) g(x) = x cos(x) Pasticulas solution form = (Ao + A1 x + Azz²+ Az x3, sinx +(Ao +Ai'z + A₂' x² + Ag'x3) cosx term Here, Ao , A ,, A2, A3, Aol, Ai! A2' Aj' are constants d) g(x) = cos(x) + COS (24) + C08 (32) Pasticulas solution form for g(x) , we can consides, for every A sin (nx) + B C08 (1x) Here, for coox , y p= Asins & B CO8x -ci tor yp = Az 8in (24) it B2 20 8(2x) - (ii) for yp= Az sin 6x) + B₂ C08 (32) - (iii) so, Pasticulas solution is sum of (i), (ii) & (iii). 108 28, Cos 3x,