MthSc 208, Spring 2011 (Differential Equations)Dr. MacauleyHW 13Due Friday March 18th, 2011(1) Solve the following differential equations:(a)y00+ 6y0+ 9y= 5(b)y00=-ω2y(c)y0+ 2y=et(d)y0+ 3y= 0.(2) Find the Laplace transform of the following functions by explicitly computingZ∞0f(t)e-stdt.(a)f(t) = 3(b)f(t) =e3t(c)f(t) = cos 2t(d)f(t) =te2t(e)f(t) =e-3tsin 2t(3) Sketch each of the following piecewise defined functions, and compute their Laplace trans-forms.(a)f(t) =0,0≤t <45,t≥4(b)f(t) =t,0≤t <33,t≥3(4) Engineers frequently use theHeavyside function, defined byH(t) =0,t <01,t≥0to emulate turning on a switch at a certain instance in time.Sketch the graph of thefunctionx(t) =e0.2tand compute its Laplace transform,X(s). On a different set of axes,sketch the graph ofy(t) =H(t-3)e0.2tand calculate its Laplace transform,
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