Chapter 8: Q8-3-17P (page 404)

Find the general solution of (1.3)for an RCcircuit, with $V = V_{0} c o s ω t$ .

Short Answer

Expert verified

The general solution of 1.3 for an RC circuit with $V = V_{0} \cos ω t$ is $q = \frac{V_{\circ}}{ω^{2} R^{2} C^{2} + 1} (ω R C^{2} \sin ω t + C \cos ω t) + α e^{- t / R C}$ :

Step by step solution

Define the First-order differential equation

The linear differential equation is defined by $x' + Px = Q$ , whereP and Qare numeric constants or functions in x. It is made up of a yand a yderivative. The differential equation is called the first-order linear differential equation because it is a first-order differentiation.

Given parameter

Given equation1.2:

$L \frac{d^{2} I}{d t^{2}} + R \frac{d I}{d t} + \frac{I}{C} = \frac{d V}{d t}$

Also given RI circuit $\frac{1}{C} = 0$ with $V = V_{0} \cos ω t (ω = c o n s \tan t)$

Find the differential equation and write it in the formx'+Px=Q

From $V = V_{0} \cos ω t$ , the equation 1.3 will become

$R \frac{d q}{d t} + \frac{q}{C} = V_{0} \cos ω t$

Then the equation in the form will be

$\frac{d q}{d t} + \frac{q}{R C} = \frac{V_{0}}{R} \cos ω t$

From the equation 3.4,

$g = \int \frac{1}{R C} d t = \frac{t}{R C} e^{g} = e^{\frac{t}{R C}}$

Find the general solution of the differential equation

From the equation 3.9 ,

${qe}^{g} =∫ (\frac{V_{ο}}{R} cosωt) e^{t/RC} dt = \frac{V_{\circ}}{R} \frac{e^{- t / R C}}{ω^{2} + (1 / R C^{2})} (ω \sin ω t + \frac{1}{R C} \cos ω t) + α q = \frac{V_{\circ}}{ω^{2} R^{2} C^{2}} (ω R C^{2} \sin ω t + \cos ω t) + α e^{- t / R C}$

So, the general solution will be:

$q = \frac{V_{} \circ}{ω^{2} R^{2} C^{2} + 1} (ω R C^{2} \sin ω t + C \cos ω t) + α e^{- t / R C}$

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Find the general solution of (1.3)for an RCcircuit, with $V = V_{0} c o s ω t$ .

Short Answer

Step by step solution

Define the First-order differential equation

Given parameter

Find the differential equation and write it in the formx'+Px=Q

Find the general solution of the differential equation

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Find the general solution of (1.3)for an RCcircuit, withV=V0cosωt.

Short Answer

Step by step solution

Define the First-order differential equation

Given parameter

Find the differential equation and write it in the formx'+Px=Q

Find the general solution of the differential equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Engineering Physics

Translational Dynamics

Nuclear Physics

Magnetism and Electromagnetic Induction

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Find the general solution of (1.3)for an RCcircuit, with $V = V_{0} c o s ω t$ .