2nd-Order (Linear?) Non-Homogeneous ODE, Two Point Boundary Value

In summary, the conversation discusses finding the solution to a two-point boundary value problem involving a nonlinear ODE with a term containing the exponential function. The person is having trouble solving it analytically and is considering using Variation of Parameters. However, it is mentioned that not every ODE can be solved analytically and that using Wolfram Alpha results in a complicated solution. The suggestion is made to test the method with a simpler problem instead.
  • #1
brru25
29
0

Homework Statement



Find the solution to the two-point boundary value problem u'' + 4u' + exu = sin(8x)

with u(-1) = u(1) = 0.

Homework Equations


The Attempt at a Solution



I haven't taken an ODE course in years but I need to verify that my numerical solution to the ODE is accurate to the actual solution. The only way to do that is to find the solution to the ODE analytically, which I'm having trouble with. My only guess is to use Variation of Parameters but I can't remember how to use it at all.
 
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  • #2
You can't solve every ode analytically. And yes, the e^x*u term makes it nonlinear. Which probably makes it hard. Just for fun, I plugged into Wolfram Alpha and it gives a completely intractable 'solution' involving definite integrals of 4th order bessel functions. That's not going to be useful to you. You can get a nice, but messy, solution if you leave the e^x out. Can you test your method with that, or maybe something even easier and nicer instead?
 

FAQ: 2nd-Order (Linear?) Non-Homogeneous ODE, Two Point Boundary Value

What is a 2nd-Order Non-Homogeneous ODE?

A 2nd-Order Non-Homogeneous ODE (Ordinary Differential Equation) is a mathematical equation that involves a function and its derivatives up to the second order. It is called non-homogeneous because it contains a non-zero term on one side of the equation, unlike a homogeneous ODE where the right side is equal to zero.

What does it mean to have a Two Point Boundary Value?

A Two Point Boundary Value refers to a type of boundary condition in which the values of the function and its derivatives are specified at two different points. This is in contrast to a One Point Boundary Value, where only one point is specified. Two Point Boundary Value problems are commonly encountered in physics and engineering applications.

How do you solve a 2nd-Order Non-Homogeneous ODE with Two Point Boundary Value?

To solve a 2nd-Order Non-Homogeneous ODE with Two Point Boundary Value, you can use various methods such as the Method of Undetermined Coefficients, Variation of Parameters, or Laplace Transform. It is important to first convert the ODE into standard form before applying any of these methods.

What is the difference between a Linear and Non-Linear 2nd-Order Non-Homogeneous ODE?

The difference between a Linear and Non-Linear 2nd-Order Non-Homogeneous ODE lies in the presence of non-linear terms. A Linear ODE has all terms involving the function and its derivatives raised to the first power, while a Non-Linear ODE may have higher powers or products of these terms. The methods of solving these two types of ODEs are different.

What are some real-life applications of 2nd-Order Non-Homogeneous ODEs with Two Point Boundary Value?

2nd-Order Non-Homogeneous ODEs with Two Point Boundary Value are commonly used in various fields such as physics, engineering, and economics. They can be used to model the motion of a pendulum, the growth of a population, the displacement of a beam under a load, or the spread of disease. They are also used in control systems and signal processing.

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