Question

Differential equations. Please answer all parts of the question!

1.Consider the linear second-order ODE +2y 0. (A) What is the "characteristic polynomial"? (B) What is the "characteristic equation"? And what are the roots? (C) What is the general solution to the ODE? 2.Find the general solution to 324u-y

Answer 1

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"ODE, d^2y/dt^2 + 3 dy/dt + 2y = 0 Let y = e^rt is a solution to the ODE r = constant Rightarrow y' = r e^rt Rightarrow y"" = r^2 e^rt so r^2 e^rt + 3 re^rt + 2 e^rt = 0 Rightarrow r^2 + 3r + 2 = 0 [e^rt notequalto 0] (A) Hence the characteristic polynomial is p(r) = r^2 + 3r + 2 (B) The characteristic equation is r^2 + 3r + 2 = 0 Rightarrow (r + 2)(r + 1) = 0 Rightarrow r = -2, - 1 So roots of the characteristic equation are - 2, - 1 (c) Hence the general solution to the ODE is y(t) = c_1 e^- 2t + c_2 e^- t c_1 c_2 are arbitrary constant (2) 3 d^2y/dx^2 - 2 dy/dx - y = 0 characteristic equation is 3r^2 - 2r - 1 = 0 Rightarrow (r - 1) (3r + 1) = 0 Rightarrow r = 1, -1/3 Hence general solution is given by y(x) = c_1 e^x +c_2 e^- x/3 where e_1 e_2 are arbitrary constant"