Question

Problem 1.Consider the harmonically forced undamped oscillator described by the following ODE:mx′′+kx=F0cosωt, k >0, m >0, ω >0, F0∈R.

Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and x'(0) = 0. Show that its unique solution can be written in the form X(t) = C(sinwit)(sin wat), C = with the constant C given by 2F0 m(wă – w2) where we = Vk/m. Give expressions for the two frequencies wi and W2 in ).

Answer 1

Q1. Sok Given data: mx + &x = to cosut, - ko, no, w OFER. The given ODE can be written as x + ku= to cos(uf) - . (2) Given that werk Characteristic polynomial, pin) = ++ The roots of p(l) are & K i dr t wei, where worth Now, the complementary solution to ean @ is given te(t) = G cos (coot) + Gein (Wot) - Now, to find the particular solution, & (t), op - 1 / 4 ( to coswt) where D= d. o and D² = 2 Replace or by wh, we get & -to coslot) , where we at the cosluh) @ m100 Hence, the general solution to ear ② is given by alt) = + (t) + 4(t) = G cos wat + G sin wat toto

(6) IVP: So x(0)= 0 and (0) = 0 G coso + C sinot to coso o at tao. -> 4+ [ cos 0 1 Sinozo 6] → 9 -ws) Again n' (t) = - Wo G sin wot + wall cos wat wil so sin ut m1wowe) Atto, - wo sim Otwo G cos o - wto sino no 6,2-60% -) Wo -) 6 = 0 Thus, the general solution given by 6 becomes r(t) = - to coswot & to cos wf m2602-602) *m(23-24) [esok-es ost] (2 min (stante con ( ) - C sin ( 0 + 0 oht ein (12-0) - This is the unique solk, where ca 26

Comparing en ④ with a(t) = (sin (wit) sin ust) we get wy = w two and. Wq = www.

The problem is solved and provided above.

Here, I've used a standard result to find the particular solution, i.e. $\phi_p=\frac{1}{f(D)} f(t)$ where $F(D)= D^2+\omega_0, D= \frac{d}{dt}$ and , otherwise it'll be quite lengthy.

Fo f(t) = — cos(wt) m

Method of variation of parameters can also be used to find it, and result will be same.

Thank you.