Solving a 2nd order D.E. of a forced LRC circuit

In summary, the conversation is about solving a forced second order differential equation for a LRC circuit with given initial conditions. The person has calculated the general solution for the homogeneous equation, but is having trouble finding the particular solution using undetermined coefficients. Another person suggests using A cos(t) + B sin(t) as the particular solution and finding the constants A and B, and then matching the general solution to the initial conditions.
  • #1
zedmondson033
2
0

Homework Statement


Solve the following forced D.E. (Show work)

L=10 R=20 C=0.01 x(0)=10 x'(0)=0

Homework Equations


This is the second order D.E. for a forced LRC circuit

L(d2x/dt2)+R(dx/dt)+x/C=200sin(t)


The Attempt at a Solution


y=ygeneral+yparticular

I calculated ygeneral to be ygeneral = e-tC1cos(3t)+e-tC2sin(3t)

Now I need to find the particular solution for the DE. I've been trying to use undetermined coefficients, choosing Asin(t)+Bcos(t) as my initial guess but the algebra doesn't seem to work out. Can anyone help me out? Thanks for your time!
 
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  • #2
You'll need to find the particular solution before you can plug in the initial conditions to find the constants.
 
  • #3
Yeah, the previous problem was to find the solution to the same equation, just without the forcing function at the end, so it was homogeneous. That's how I got the values for the constants and the general equation.
 
  • #4
zedmondson033 said:
I calculated ygeneral to be ygeneral = 10e-tcos(3t)+(10/3)e-tsin(3t)
This is not the general solution of the homogeneous equation: It has to contain two undetermined constants.

The "force" function has different frequency from the own frequency of the LCR circuit: so the particular solution is in the form A cos(t) +Bsin(t), as you supposed. Find A and B. Show what you have tried.
Match the general solution x(homogeneous) +x(particular) to the initial conditions.

ehild
 

1. What is a 2nd order D.E. in relation to a forced LRC circuit?

A second order differential equation (D.E.) is a mathematical equation that involves the second derivative of a function and is commonly used to describe dynamic systems. In the context of a forced LRC circuit, a second order D.E. is used to model the behavior of the circuit over time, taking into account the effects of the inductor (L), resistor (R), and capacitor (C) components.

2. How is a forced LRC circuit different from an unforced LRC circuit?

An unforced LRC circuit is one in which the input voltage is constant and does not vary over time. In contrast, a forced LRC circuit has an input voltage that varies over time, often in the form of an external signal or forcing function. This variation in input voltage leads to a more complex behavior in the circuit, which is why a second order D.E. is needed to model it.

3. What are the steps for solving a 2nd order D.E. of a forced LRC circuit?

The first step is to write out the differential equation, which will involve the second derivative of the current in the circuit. Then, apply any necessary simplifications or substitutions to make the equation more manageable. The next step is to find the general solution to the differential equation by using techniques such as separation of variables or the method of undetermined coefficients. Finally, apply any initial conditions or boundary conditions to find the specific solution for the circuit.

4. Can the solution to a 2nd order D.E. of a forced LRC circuit be verified experimentally?

Yes, the solution to a 2nd order D.E. of a forced LRC circuit can be verified experimentally by comparing the predicted behavior of the circuit to actual measurements. This can be done by building a physical circuit with the same components and input voltage as the one described in the D.E., and then measuring the current over time and comparing it to the calculated solution.

5. Are there any real-world applications of solving a 2nd order D.E. of a forced LRC circuit?

Yes, there are many real-world applications of solving a 2nd order D.E. of a forced LRC circuit. One example is in the design of electronic filters, which are used to selectively pass or block certain frequencies in a circuit. By solving the D.E. for a forced LRC circuit, engineers can design filters that have the desired frequency response for a specific application, such as in audio systems or communication systems.

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