(a) Find the general solution (2) of the homogeneous differential equation where m and k are positive constants. The c's below denote arbitrary constants. Hint: This is an Homogeneous SOLDE, find the general solution in the usual way. k A. ₁ (t)=₁+₂et, with w=. V m cos(wt) + ₂ sin(wt), with w= B. 3/h (t) = c. ₁ (t) = ₁ + 0₂ et, with w= k m k D. Yk (t) = q et + c₂tet, with W= k E. 3 (t)= c₂ cos(wt) + ₂ sin(at), with @= Vm k F. yh(t) = ₂ et + c₂tet, with w= Vm G.y₁ (t) = C₁ sin(wt), with w= O. H. yh(t)= c₂ cos(wt), with w OI. None of the above. m ▾ Part 2: Non-Homogeneous Equation: Non-Resonant (b) Let w be as found in the solution of part (a). Find the solution yr (t) of the non-homogeneous initial value problem = A. Ynr(t) = (cos(vt) sin(wt)) ⒸB. Ymr (t) = 1 (wi²-2²) 1 (w²-2²) 1 (cos(vt) - cos(wt)) OC. Ymr (t) = (²-²) (cos(vt) + cos(wt)) 1 OD. Ymr (t) = (²-2) (sin(vt) – sin(wt)) - OE. Yur(t): (sin(vt) + sin(wt)) 1 (w²2 +2²) 1 (cos(vt) + cos(wt)) (cos(vt) - cos(wt)) (sin(vt) - sin(wt)) (sin(vt) sin(wt)) OF. Ymr (t) = OG. Ynr(t) = OH. Your(t) = (w² +2²) 1 1. Yor(t) = (²2²) OJ. None of the above. V m k m (w² +2²) 1 Hint: You need to do many things here: 1) Solve the homogeneous equation. 2) Find a particular solution of the non-homogeneous equation (for example by guessing, here the first guess is ok). See Example 2.3.5. 3) Use 1) and 2) to write the general solution of the non-homogeneous equation. 4) Use the initial conditions to find the solution of the initial value problem. (w² +2²) i k m my" + ky = 0, y"+w²y = cos(vt), y(0) = 0, y'(0) = 0, with v‡w.