Final answer:
The second-order linear homogeneous differential equation y''+196y=0 can be solved by assuming solutions of the form y = e^rt, leading to the general solution y(t) = C1 cos(14t) + C2 sin(14t), with C1 and C2 as arbitrary constants.
Step-by-step explanation:
The question requires solving the second-order linear homogeneous differential equation y''+196y=0. This is a classical differential equation problem often encountered in fields like physics and engineering. To solve it, we look for solutions of the form y = e^rt where r is a constant.
Substituting y = e^rt in the differential equation, we get r^2e^rt + 196e^rt = 0. Therefore, e^rt(r^2 + 196) = 0. Since e^rt never equals zero, we set the quadratic equation r^2 + 196 = 0 to find the roots. Solving for r yields r = ±14i, indicating a pair of complex conjugate roots, r= 14i and r= -14i. The general solution to the differential equation then is y(t) = C1 cos(14t) + C2 sin(14t) where C1 and C2 are arbitrary constants determined by initial conditions.