. Utilizing Laplace transforms and matrix-vector formulation, solve the initial value problem yi = -2y2 + 5 sin t · u(t – 2T), y2 = -2y, with y,(0) = 0 and y2(0) = 1. The following partial fraction expansion might be useful in finding inverse Laplace transforms. As + B Cs + D – a² ' s² + w² 1 %3D (s² – a²)(s² + w²) ¯ s² where the coefficients A, B, C and D can be easily determined by considering s | = jw.

(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

Solve the initial value problem below using the method of Laplace transforms. w'' - 4w' + 4w=24t+28, w(-2)= -2, w'(-2) = 5 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms.