Question

Linear Algebra and differential Equations.

Consider the RLC circuit with R = 180Ω, C = 1/280F, L = 20H, and applied voltage E(t) = 10 sin t. Assuming no initial charge on the capacitor, but an initial current of 1 ampere. Determine the charge on the capacitor for t> 0. (a) Write the differential equation, y', +4xy'-6x2y = x2 sin x, as an operator equation and give the associated homogeneous DE (b) Write the DE (D2 1) (D 3)(y) e In(2 1) in terms of the expressions y, y', y", etc. Is this differential equation linear? Why or why not

Answer 1

The differential Equation corresponding to ELC circuit is L Q'' + RQ' + 1/c Q = E(+) ---(1) Rightarrow 20 Q'' + 180 Q' + 280 Q = 10 sin t ---(2) Q(0) = 0, Q'(0) = 1 which can be rewrite as (20 D^2 + 180 + 280) Q = 10 sin t ---(3) (D = d/dr, 13^2 = d^2/dx^2 The Auxiliary equation is 20m^2 + 180 + 280 = 0 Rightarrow m^2 + 9D + 14 = 0 Rightarrow (m + 7) (m + 2) = 0 Rightarrow m = -7, - 2 The solution for homogenous part, complementary solution is Q_c = c_1 e^- 7t + c_2 e^- 2t the particular solution is Q_p Q_p = 1/20 D^2 + 180 D + 280 10 sin t = 10 1/20 (-1^2) + 180 D + 280 sin t(D^2 = -1^2) = 10/180 D + 260 sin t = sin t/18 D + 26 = 1/2 1/90 + 13 sin t