Problem 3. A connection to second-onter ODE Recall the second-order ODE ay" + by' cy g(t), where ...
What is the general intuition behind picking a PARTICULAR solution to a second-order, linear, non-homogenous, ODE y′′+p(t)y′+q(t)y=g(t) instead of following the rules for example when seeing an exponential you know the guess has to include an exponential? P.S I heard the intuition follow somehow checking your guess with the ODE on the left side of the equation and its derivatives before actually applying it, but how?
Consider the following statements. (i) Given a second-order linear ODE, the method of variation of parameters gives a particular solution in terms of an integral provided y1 and y2 can be found. (ii) The Laplace Transform is an integral transform that turns the problem of solving constant coefficient ODEs into an algebraic problem. This transform is particularly useful when it comes to studying problems arising in applications where the forcing function in the ODE is piece-wise continuous but not necessarily...
+ (3) ar2 2. Recall from lectures that the governing PDE for vibrations of a circular drum lid is 1 au 1 ay c? + 012 72 302 for r € (0,R), 0€ (-2,7), and t > 0, and the boundary condition is (R, 6,t) = 0 for t>0 and -150<7. rar (4) You will search for a solution of the form v(r,0,t) = G(r) sin(30) cos(w t), (5) for a function G that satisfies the ODE m2 G" +rG'...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...