Chapter 1: Q8-3-16P (page 1)

Find the general solution of(1.2)for an RLcircuit $(\frac{1}{C} = 0)$ with $V = V_{\circ} c o s ω t (ω = c o n s t a n t)$ .

Short Answer

Expert verified

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

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 equation 1.2:

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

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 $V = V_{0} \cos ω t (ω = c o n s \tan t)$ , the equation 1.2 will become

$L \frac{d I}{d t} + R I = V_{0} \cos ω t$

Then the equation in the form will be

$\frac{d I}{d t} + \frac{R}{L} I = \frac{V_{0}}{L} \cos ω t$

From the equation 3.4,

$f = \int \frac{R}{L} d t = \frac{R}{L} t e^{f} = e^{\frac{R t}{L}}$

Find the general solution of the differential equation

From the equation 3.9,

${le}^{f} =∫ (\frac{V_{\circ}}{L} cosωt) e^{Rt/L} dt = \frac{V_{\circ}}{L} \frac{e^{- R t / L}}{ω^{2} + (R / L^{2})} (ω \sin ω t + \frac{R}{L} \cos ω t) + α I = \frac{V_{\circ}}{ω^{2} L^{2} + R^{2}} (ω L \sin ω t + R \cos ω t) + α e^{- R t / L}$

So, the general solution will be:

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

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Find the general solution of(1.2)for an RLcircuit $(\frac{1}{C} = 0)$ with $V = V_{\circ} c o s ω t (ω = c o n s t a n t)$ .

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(1.2)for an RLcircuit(1C=0)withV=V∘cosωt(ω=constant).

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

Linear Momentum

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Study anywhere. Anytime. Across all devices.

Find the general solution of(1.2)for an RLcircuit $(\frac{1}{C} = 0)$ with $V = V_{\circ} c o s ω t (ω = c o n s t a n t)$ .