Solving a Differential Equation with L, R, E(t), and i(0) as Constants

In summary, the conversation is discussing how to solve a differential equation with given constants and initial conditions. The method involves using an integrating factor and then integrating using either parts or complex analysis. The final solution may be messy.
  • #1
hola
38
0
I am stumped... here is the problem:
Solve the DE using the following:
L and R are constants

[tex]L\frac{di}{dt} + Ri = E(t)[/tex]

[tex]i(0) = i_0[/tex]

[tex]E(t) = E_0*sin(wt)[/tex]

Here is my work so far:

I got the integrating factor to become [tex]e^{Rt/L}[/tex]. But now:

[tex]\frac{d(e^{\frac{Rt}{L}}*i)}{dt} = e^{\frac{Rt}{L}}\frac{E_0}{L}*sin(wt)[/tex]

But I am stuck from there. Help would be appreciated.
 
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  • #2
Since you have

[tex]\mu(t) = e^{\frac {R}{L}t}[/tex]

and you also have

[tex](\mu(t)i)' = \mu(t) \frac {E_0}{L} \sin (\omega t)[/tex]

You can integrate both sides and divide by [itex]\mu(t)[/itex]

Hence
[tex]i(t) = \frac {\int \mu(s) \frac {E_0}{L} \sin (\omega s) ds}{\mu(t)}[/tex]

I switched the t to a s in the numerator to avoid confusion. After you integrate the numerator, you can replace the s with a t.
 
  • #3
That's the problem... I can't integrate it.
 
  • #4
Integrate it by parts. Let [itex]u=\sin(\omega t)[/itex] and let [itex]dv=exp\left(\frac{Rt}{L})[/itex].

You'll have to integrate by parts twice and then algebraically solve for the integral. This integral actually pops up all the time in second order dynamic systems.
 
  • #5
You can either integrate by parts or, if you're comfortable with complex analysis, you can note that

[tex]\int e^{at} \sin \omega t dt = I am \int e^{(a + i \omega) t} dt[/tex]

and extract the imaginary part after performing the integration.
 
  • #6
I got some really messy answer... is that ok?
 
  • #7
That depends on the answer! :biggrin:

Why don't you post what you did so we can see it?
 

Question 1: What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model and solve many physical, natural, and social phenomena.

Question 2: What are L, R, E(t), and i(0) in a differential equation?

L, R, E(t), and i(0) are constants that represent the inductance, resistance, applied voltage, and initial current, respectively, in an electrical circuit. They are used as parameters in the differential equation to solve for the current as a function of time.

Question 3: How do you solve a differential equation with L, R, E(t), and i(0) as constants?

To solve a differential equation with these constants, you can use techniques such as separation of variables, substitution, or integrating factors. These methods involve manipulating the equation to isolate the dependent variable and then finding a solution that satisfies the initial conditions.

Question 4: What is the significance of solving a differential equation with L, R, E(t), and i(0) as constants?

Solving a differential equation with these constants allows us to understand and predict the behavior of an electrical circuit. By finding the current as a function of time, we can analyze the changes in current over time and make informed decisions about the design and operation of the circuit.

Question 5: What are some real-world applications of solving a differential equation with L, R, E(t), and i(0) as constants?

Solving differential equations with these constants has many practical applications, such as in designing electrical circuits, predicting the growth of populations, modeling chemical reactions, and analyzing the spread of diseases. It is also used in fields such as engineering, physics, biology, and economics to understand and solve complex systems.

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