Chapter Contents

• Differential Equations
• Predicting AIDS - a DEs example
• 1. Solving Differential Equations
• 2. Separation of Variables
• 3. Integrable Combinations
• 4. Linear DEs of Order 1
• 5. Application: RL Circuits
• 6. Application: RC Circuits
• 7. Second Order DEs - Homogeneous
• 8. Second Order DEs - Damping - RLC
• 9. Second Order DEs - Forced Response
• 10. Second Order DEs - Solve Using SNB
• Differential Equations Problem Solver

• Comments, Questions?

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From the math blog...

9. The Forced Response - Second Order Linear DEs

As in first order circuits, the forced response has the form of the driving function. For a constant driving source, it results in a constant forced response.

For non-constant driving functions e.g. when

E = E₀ sin ωt,

the complete response of a circuit is the sum of a natural response and a forced response.

Note: Such solutions can also be obtained using the Laplace transformation method (which we meet later) when initial conditions are given.

Constant Forced Response

Example

In a RCL series circuit, R = 10 Ω, C = 0.02 F, L = 1 H and the voltage source is E = 100 V. Solve for the current i(t) in the circuit given that at time t = 0, the current in the circuit is zero and the charge in the capacitor is 0.1 C.

Answer

Natural and Non-Constant Forced Response

Example 6 (Using Scientific Notebook)

Here is an example using Scientific Notebook showing the effect of a forced response. (You need SNB to see the result. See SNB information).

a. Natural Response

In an RLC circuit we have L = 1 H , R = 10 Ω and C = 0.0025 F and at t = 0, the current is 0 and i'(0) = 0.1 A/s.

Solve for i.

Solution:

We need to solve:

[In Scientific Notebook, go to Compute menu, Solve ODE..., Exact]

Get the SNB file.

b. Forced Response

Let us now take the RLC circuit in Example 6a but have a non-constant EMF

E = -0.08 cos 2.5t

Using the result (from Section 8)

with L = 1, R = 10 and 1/C = 400, we differentiate throughout with respect to t to give the following 2nd order DE, with initial conditions shown:

See the SNB file (.TEX file) for the (long, long) result...

[Go here for SNB information.]

Now for the graphs that we obtain using our SNB solution:

 

We will also use the Laplace Transform in a later section to solve this type of DE.