Homework 03 @ 2021-02-06 00:53:46-08:00EECS 16BDesigning Information Devices and Systems IISpring 2021UC BerkeleyHomework 03This homework is due on Friday, February 5, 2021, at 11:00PM. Self-gradesand HW Resubmission are due on Tuesday, February 9, 2021, at 11:00PM.1. Reading Lecture NotesStaying up to date with lectures is an important part of the learning process in this course. Here are links tothe notes that you need to read for this week:Note 2andNote 3A(a) How do we deal with piecewise constant inputs as introduced in the notes?

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(b) What conclusions do we get after approximating any functionu(t)as being piecewise constant overfixed interval widthsΔ?

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2. Simple scalar differential equations driven by an inputIn class, you learned that the solution fort≥0 to the simple scalar first-order differential equationddtx(t) =λx(t)(1)with initial conditionx(t=0) =x0(2)is given fort≥0 byx(t) =x0eλt.(3)In an earlier homework, you proved that these solutions areunique– that is, thatx(t)of the form in (3) arethe only possible solutions to the equation (1) with the specified initial condition (2).In this question, we will analyze differential equations with inputs and prove that their solutions are unique.In particular, we consider the scalar differential equationddtx(t) =λx(t)+u(t)(4)whereu(t)is a known function of time fromt=0 onwards.(a) Suppose that you are given anxg(t)that satisfies both (2) and (4) fort≥0.Show that ify(t)also satisfies(2)and(4)fort≥0, then it must be thaty(t) =xg(t)for allt≥0.(HINT: You already used ratios in an earlier HW to prove that two things were necessarily equal. Thistime, you might want to use differences. Be sure to leverage what you already proved earlier insteadof having to redo all that work.)EECS 16B, Spring 2021, Homework 031