PP6 **Transcription for Educational Website****Problem Statement:**6. Show that \( q(t) = q_0 \exp(-t/RC) \) is a solution of the ordinary differential equation (ODE): \[ Ri(t) + \frac{q(t)}{C} = 0 \]**Explanation:**This problem involves showing that the given function \( q(t) = q_0 \exp(-t/RC) \) satisfies the provided ordinary differential equation. The ODE is a first-order linear differential equation commonly seen in the context of electrical circuits, specifically RC (resistor-capacitor) circuits. We are asked to verify if the exponential function, which represents the charge \( q \) over time \( t \), is a solution by substituting it into the equation.**Notes:**- \( q(t) \) denotes the charge as a function of time.- \( q_0 \) is the initial charge.- \( \exp \) refers to the exponential function.- \( R \) is the resistance.- \( C \) is the capacitance.- \( i(t) \) implies the current as a function of time.**Application:**To verify, differentiate \( q(t) \) with respect to time \( t \), relate it to the current function \( i(t) \), and substitute into the ODE to check if it holds true. This problem highlights how exponential decay models the discharge of a capacitor in an RC circuit.

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6. Show that q(t) = qoexp(-t/RC) is a solution of the ordinary differential equation (ODE): Ri(t)+q(t)/C = 0

Physics for Scientists and Engineers with Modern Physics

10th Edition

ISBN:9781337553292

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter38: Relativity

Section: Chapter Questions

Problem 49AP

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Question

PP6

**Transcription for Educational Website**

**Problem Statement:**

6. Show that \( q(t) = q_0 \exp(-t/RC) \) is a solution of the ordinary differential equation (ODE):

\[ Ri(t) + \frac{q(t)}{C} = 0 \]

**Explanation:**

This problem involves showing that the given function \( q(t) = q_0 \exp(-t/RC) \) satisfies the provided ordinary differential equation. The ODE is a first-order linear differential equation commonly seen in the context of electrical circuits, specifically RC (resistor-capacitor) circuits. We are asked to verify if the exponential function, which represents the charge \( q \) over time \( t \), is a solution by substituting it into the equation.

**Notes:**
- \( q(t) \) denotes the charge as a function of time.
- \( q_0 \) is the initial charge.
- \( \exp \) refers to the exponential function.
- \( R \) is the resistance.
- \( C \) is the capacitance.
- \( i(t) \) implies the current as a function of time.

**Application:**

To verify, differentiate \( q(t) \) with respect to time \( t \), relate it to the current function \( i(t) \), and substitute into the ODE to check if it holds true. This problem highlights how exponential decay models the discharge of a capacitor in an RC circuit.

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