Question

differential equations. PLEASE answer both parts. These have to do with Superposition principles. thank you in advance. Will upvote!

Find the general solution to y" + y, + y = et +2e-t. Find the general solution to y +4y (t +2)e t.

Answer 1

"y"" + y' + y = e^t + 2e^-t y"" + y' + y = 0 rightarrow (1) y"" + y' + y = e^t rightarrow (2) y"" + y' + y = e^-t rightarrow (3) solution for (1), m_2 + m + 1 = 0 [auxiliary equation] m = -1 plusminus squareroot 1 - 4.1.1/2.1 = -1 plusminus squareroot 3 i/2 = -1/2 plusminus squareroot 3/2 i the solution is y_t = e^-1/2t (near squareroot 3/2 t + B sin squareroot 3/2 t) rightarrow complementary function [A and B are arbitrary constant] Now We will find the particular integral. [general solution for a ODE is = complementary function + particular integral] particular integral for (2) is y_p = 1/D^2 + D + 1 (e^t) [D = d/dt] = e^t/1^2 + 1 + 1 = 1/3 e^t particular integral for (3) is, y_p = 1/D^2 + D + 1 e^-t = e^-t/(-1)^2 + (-1) + 1 = e^-t particular integral for, y"" + y' + y = e^t + 2e^-t"