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For each of the following, find a differential equation with the given solution. (Hint: Use what you know about characteristic polynomials) (a) y(t) = c1e−t + c2e−4t (b) P(t) = c1e2t + c2te2t (c) y(t) = c1cos⁡(5t) + c2sin⁡(5t) (d) P(t) = c1et + c2e−2t + c3e6t

For each of the following, find a differential equation with the given solution. (Hint: Use what you know about characteristic polynomials) (a) y(t) = c1e−t + c2e−4t (b) P(t) = c1e2t + c2te2t (c) y(t) = c1cos⁡(5t) + c2sin⁡(5t) (d) P(t) = c1et + c2e−2t + c3e6t

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For each of the following, find a differential equation with the given solution. (Hint: Use what you know about characteristic polynomials) (a) y ( t ) = c 1 e − t + c 2 e − 4 t (b) P ( t ) = c 1 e 2 t + c 2 t e 2 t (c) y ( t ) = c 1 cos ⁡ ( 5 t ) + c 2 sin ⁡ ( 5 t ) (d) P ( t ) = c 1 e t + c 2 e − 2 t + c 3 e 6 t

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For an NMOS differential pair with a common-mode voltage VCM applied, as shown in Fig. 9.2, let VDD = VSS = 1.0 V, kn' = 0.4 mA/V^2, (W/L) 1,2 = 10, Vtn = 0.4 V, I = 0.16 mA, RD = 5kohm, and neglect channel-length modulation. (a) Find VOV and V GS for each transistor. (b) For V CM =0, find VS, ID1, ID2, VD1, VD2, and VO. (c) Repeat (b) for VCM = +0.4 V. (d) Repeat (b) for VCM = -0.1 V (e) What is the highest value of VCM for which Q1 and Q2 remain in saturation? (f) If current source I requires a minimum voltage of 0.2 V to operate properly, what is the lowest value allowed for VS and hence for VCM (g) What is the input common-mode range?
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