Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function -4-5(t-1), (0)-6, (0) 0. = a Find the Laplace transform of the solution Y()-C (v(t)}- b. Obtain the solution y(t). (1) c Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t= 1. if 0 < t < 1, {₁ y(t) = if 1 ≤ t < 00.

(5) Find the transfer function h(@)= y(@)/x(@) of the following ODE and plot its magnitude. y"(t)+y'(t) + y(t) = x(t) HINT: Take the Fourier Transform of the ODE to obtain - a²ŷ(0) + iaỹ(a) + y(@) = x(w) Giving h(o)= = y(@) 1 x(@) - @²+ io +1 The magnitude of the transfer function is then |ħ(w) = 1 1- @²} + @² |h(@) 0.5 0.8 0.7 0.6 0.5 0.4 0.3 e 0.2 0,4 0.6 0.8 (10) e 1 1.2 1.4 1.6 1.8 2

An equation of the form +2d²y dt2 dy +at dt + By = 0, t> 0, where a and ẞ are real constants, is called an Euler equation. α (a). Let x = Int and calculate dy/dt and d²y/dt² in terms of dy/dx and d²y/dx². (b) Show that one can use the results of part (a) to transform the original equation into d²y dy + (α − 1). dx² + By = 0. dx Observe now that the resulting differential equation has constant coefficients. (c) Show that if y₁(x) and y2(x) form a fundamental set of solutions of the latter equation in part (b), then y₁ (Int) and y2 (Int) form a fundamental set of solutions of the original equation. (d) Using all above observations to solve 1²y" + 4ty' + 2y = 0