The ODE below models a circuit with inductance L, resistance R, capacitance Cand voltage source E(t).

 0.0008i''+1000i'+(i/0.0007)=cos(180t)


Find the constants and voltage source function.

 L= 
 R=
C= 
  E(t)= 


To find the constants and the voltage source function, we need to compare the given differential equation with the standard form of the second-order linear differential equation for an RLC circuit with an external voltage source \( E(t) \):

\[ L i'' + R i' + \frac{1}{C} i = E(t) \]

Given equation:
\[ 0.0008i'' + 1000i' + \frac{i}{0.0007} = \cos(180t) \]

By comparing the coefficients, we can identify the constants \( L \), \( R \), \( C \), and the voltage source function \( E(t) \):

1. The coefficient of \( i'' \) is \( L \):
   \[ L = 0.0008 \, \text{H} \]

2. The coefficient of \( i' \) is \( R \):
   \[ R = 1000 \, \Omega \]

3. The coefficient of \( i \) is \( \frac{1}{C} \):
   \[ \frac{1}{C} = \frac{1}{0.0007} \]
   Solving for \( C \):
   \[ C = 0.0007 \, \text{F} \]

4. The voltage source function \( E(t) \) is the right-hand side of the equation:
   \[ E(t) = \cos(180t) \]

Therefore, the constants and the voltage source function are:

\[ L = 0.0008 \, \text{H} \]
\[ R = 1000 \, \Omega \]
\[ C = 0.0007 \, \text{F} \]
\[ E(t) = \cos(180t) \]

Would you like any details or have any questions?

Here are some related questions you can ask:
1. How do you solve this differential equation for \( i(t) \)?
2. What are the initial conditions for solving this ODE?
3. How do the values of \( L \), \( R \), and \( C \) affect the circuit's behavior?
4. Can you explain the physical significance of each term in the differential equation?
5. How can the resonance frequency of this RLC circuit be calculated?

**Tip:** In RLC circuits, the resonance frequency is given by \( \omega_0 = \frac{1}{\sqrt{LC}} \). This is the frequency at which the circuit naturally oscillates when not driven by an external source.

Learn how to solve a second-order linear differential equation modeling an RLC circuit with inductance L, resistance R, capacitance C, and voltage source E(t). Understand the constants L, R, C, and the voltage source function E(t) involved in the differential equation. Suitable for undergraduate students studying differential equations and circuit theory.

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