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Write a differential equation that meets the given criteria: (a) A second-order differential equation that we could use a characteristic polynomial to solve. (b) A second-order differential equation that we would need the method of undetermined coefficients in order to solve. (c) A second-order differential equation for which neither a characteristic polynomial nor the method of undetermined coefficients would yield a solution. (d) Would the characteristic polynomial method work if applied to a first-degree homogeneous differential equation with constant coefficients? Explain why or why not.

Write a differential equation that meets the given criteria: (a) A second-order differential equation that we could use a characteristic polynomial to solve. (b) A second-order differential equation that we would need the method of undetermined coefficients in order to solve. (c) A second-order differential equation for which neither a characteristic polynomial nor the method of undetermined coefficients would yield a solution. (d) Would the characteristic polynomial method work if applied to a first-degree homogeneous differential equation with constant coefficients? Explain why or why not.

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Write a differential equation that meets the given criteria: (a) A second-order differential equation that we could use a characteristic polynomial to solve. (b) A second-order differential equation that we would need the method of undetermined coefficients in order to solve. (c) A second-order differential equation for which neither a characteristic polynomial nor the method of undetermined coefficients would yield a solution. (d) Would the characteristic polynomial method work if applied to a first-degree homogeneous differential equation with constant coefficients? Explain why or why not.

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The charge present in a circuit capacitor at time t is given by Q(t). The rate of change of the charge in the capacitor, dQ dt, then represents the current running through the circuit. By applying Faraday's Law, Ohm's Law, and Kirchoff's Law, we can describe the behavior of the circuit using the second-order differential equation LQ′′ + RQ′ + 1/CQ = E(t) where L is an inductance constant (in units of henrys), R is the resistance (in units of ohms), and C is the constant of capacitance for the capacitor (in units of farads). E(t) is some function in terms of t representing the voltage of the battery. Consider the case in which we have a circuit with an inductance constant of 2 henrys, a resistance of 6 ohms, and 0.25 is our constant of capacitance. The voltage of the battery can be represented by the function 8 sin⁡(2t). (a) Write out a differential equation (using the general form given above) to represent this circuit system. 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Consider the case in which we have a circuit with an inductance constant of 2 henrys, a resistance of 6 ohms, and 0.25 is our constant of capacitance. The voltage of the battery can be represented by the function 8 sin⁡(2t). (a) Write out a differential equation (using the general form given above) to represent this circuit system. (b) Solve this differential equation as though it were homogeneous (find Qh for this instance). (c) Use the Method of Undetermined Coefficients to identify the particular solution Qp for this instance. (d) What is the generalized solution to your differential equation given in part (a)? (e) Briefly explain (1-2 sentences) what the generalized solution represents in this problem's context.The charge present in a circuit capacitor at time t is given by Q(t). The rate of change of the charge in the capacitor, dQ dt, then represents the current running through the circuit. 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