Δ PROBLEM 2(25pts) The mathematics in this course centers on two topics: (i) differential equations, and (ii) polynomial

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Δ PROBLEM 2(25pts) The mathematics in this course centers on two topics: (i) differential equations, and (ii) polynomials. The goal of this problem is to get you to understand their connection. Suppose that a function of time x(t) is differentiable [i.e. dx/dt=i(t)exists], and that it has the initial condition x(t=0_)=x.. The Laplace transform of x(t) is defined as: ((x)(3) = (t)e**'dt= x(s), where s=0+im is allowed to be a complex number. 1=0. = UV- (a)(5pts) Recall from integral calculus: integration by parts: ſu cv = ? ſv du. Use this to show that, so long as lim r(t)e ** =, we have the following Laplace transform relation for x(t): [()= (x(+8*" di- **dt=s X(S)-X 100 =0 Solution: