Particular Solutions for Non-Homogeneous Differential Equations

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In summary, "Particular Sol to ODE" is a specific solution to an ordinary differential equation (ODE) that satisfies given initial conditions. It differs from the general solution, which is a family of solutions that includes all possible solutions to the equation. There are several methods for finding the particular solution, and it is unique when the ODE is linear and initial conditions are given at a specific point. Finding the particular solution is important for solving real-world problems and understanding the behavior of systems.
  • #1
dspampi
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Find a particular solution for the following non-homogeneous di eren-
tial equation by the method of undetermined coefficients:

a. y'' + 8y' +12y = e^-2x + sin(2x)

b. y'' + 11y' - 12y = 3x^2 + 4 + e^x



I got for a. Yp(x) = 1/4xe^-2x + 1/40cos(2x) +1/20 sin(2x)

b. Yp(x) = -1/4x^2 - 11/24x - 229/288 - 1/13xe^-x

I have a feeling that a is right but not sure about b.
 
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  • #2
Plug them back into the NH equation and see if they work. Then you will be sure.
 

What is "Particular Sol to ODE"?

"Particular Sol to ODE" refers to the particular solution to an ordinary differential equation (ODE). It is a specific solution that satisfies the given initial conditions for the ODE.

How is the particular solution to an ODE different from the general solution?

The general solution to an ODE is a family of solutions that includes all possible solutions to the equation. The particular solution, on the other hand, is a specific solution that satisfies the given initial conditions for the equation.

What are some methods for finding the particular solution to an ODE?

There are several methods for finding the particular solution to an ODE, including the undetermined coefficients method, variation of parameters method, and the method of annihilators.

When is the particular solution unique?

The particular solution to an ODE is unique when the ODE is linear and the initial conditions are given at a specific point. However, if the initial conditions are given over an interval, there may be multiple particular solutions.

What is the importance of finding the particular solution to an ODE?

Finding the particular solution to an ODE allows us to solve real-world problems that can be modeled using ODEs. It helps us understand the behavior of a system and make predictions based on the given initial conditions.

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