Understanding and Solving ODEs with Inhomogeneous Boundary Conditions

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In summary, the conversation is about a proof for the solution of the diffusion equation with inhomogeneous boundary conditions. The equation involves a term in the dependent variable, which the person is unsure how to handle. The book provides a solution using an integration factor.
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StewartHolmes
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I'm trying to follow a proof for the solution of the diffusion equation in 0 < x < l with inhomogeneous boundary conditions.

[tex] \frac{d u_n(t)}{dt} = k( -\lambda_n u_n(t) - \frac{2n\pi}{l}[ (-1)^n j(t) - h(t) ] )[/tex]
[tex]u_n(0) = 0[/tex]

Now I just plain don't understand what kind of an ODE I have here. If the term in j(t) and h(t) wasn't there, it'd be a simple ODE, but I'm confused as to what can be done now. I know ODEs of the form

y' + p(x)y + q(x) = 0

But I have something like, y' + p(x)y + q(t) where I have a term in the dependent variable.

The book I have gives the solution as
[tex] u_n(t) = Ce^{-\lambda_n kt} - \frac{2n\pi k}{l}\int\limits_0^t e^{-\lambda_n k(t-s)} \left( (-1)^n j(s) - h(s) \right) \, ds [/tex]
 
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  • #2
Try and learn to encapsulate everything. You have:

[tex]
\frac{d u_n(t)}{dt} = k( -\lambda_n u_n(t) - \frac{2n\pi}{l}[ (-1)^n j(t) - h(t) ] )
[/tex]

Now, isn't the term:

[tex]-\frac{2n\pi}{l}k[(-1)^n j(t)-h(t)][/tex]

just some function of t? Say it's v(t). So you have essentially the equation:

[tex]\frac{dy}{dt}+k\lambda y=v(t)[/tex]

And you know how to integrate that right by finding the integration factor. Change it to u_n if you want, but it's the same equation essentially.
 
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1. What is an ODE?

An ODE, or Ordinary Differential Equation, is an equation that contains an unknown function and its derivatives with respect to one or more independent variables. It is used to model many physical, chemical, and biological systems.

2. What are the different types of ODEs?

There are several types of ODEs, including linear, nonlinear, first-order, and second-order. Linear ODEs have a linear relationship between the unknown function and its derivatives, while nonlinear ODEs have a nonlinear relationship. First-order ODEs involve the first derivative of the unknown function, while second-order ODEs involve the second derivative.

3. How do you solve an ODE?

The method for solving an ODE depends on the type of ODE and the specific equation. Some common methods include separation of variables, substitution, and variation of parameters. Numerical methods, such as Euler's method and Runge-Kutta methods, can also be used to approximate solutions.

4. What are some real-world applications of ODEs?

ODEs are used to model a wide range of systems, including population dynamics, chemical reactions, mechanical systems, and electrical circuits. They are also used in fields such as engineering, physics, biology, and economics.

5. How do ODEs relate to PDEs?

ODEs are a type of differential equation that involve only one independent variable. PDEs, or Partial Differential Equations, involve multiple independent variables. ODEs can be seen as a special case of PDEs, where the equation does not depend on more than one independent variable.

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