2nd Order In-Homogeneous Particular Solution?

Thread starter raaznar
Start date May 26, 2013
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2nd order Particular solution

In summary, a 2nd Order In-Homogeneous Particular Solution is a type of solution to a second-order differential equation that includes a non-zero constant term. It differs from a Homogeneous Solution by including a constant term, and is used to find the specific solution to a differential equation with boundary conditions. The process for finding this solution involves setting up the differential equation, determining the homogeneous solution, and using the method of undetermined coefficients to find the particular solution. This solution can have complex roots and is commonly used in science to model physical systems and make predictions about their behavior.

May 26, 2013

raaznar

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Last edited: May 26, 2013

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May 26, 2013

HallsofIvy

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Yes, the characteristic equation is [itex]r^2+ 4r+ 3= (r+ 3)(r+ 1)= 0[/itex] which has roots r= -1 and r= -3 so the general solution to the associated homogeneous equation is [itex]Ae^{-3x}+ Be^{-x}[/itex].

For [itex]2e^{2x}cos(x)[/itex] try [itex]e^{2x}(Ccos(x)+ Dsin(x)[/itex]. For ##3xe^{-4x}##, try [itex]e^{-4x}(Ex+ F)[/itex]. For 3, try G.

Last edited by a moderator: May 26, 2013

1. What is a 2nd Order In-Homogeneous Particular Solution?

A 2nd Order In-Homogeneous Particular Solution is a type of solution to a second-order differential equation that includes a non-zero constant term. It is used to find the specific solution to a differential equation with boundary conditions.

2. How is a 2nd Order In-Homogeneous Particular Solution different from a Homogeneous Solution?

A Homogeneous Solution only includes the non-constant terms of a differential equation, while a 2nd Order In-Homogeneous Particular Solution includes a constant term. This makes the particular solution different from the general solution, as it is specific to the given boundary conditions.

3. What is the process for finding a 2nd Order In-Homogeneous Particular Solution?

The process for finding a 2nd Order In-Homogeneous Particular Solution involves setting up the differential equation with the given boundary conditions, determining the homogeneous solution, and then using the method of undetermined coefficients to find the particular solution. This involves guessing a particular solution and plugging it into the differential equation to solve for the coefficients.

4. Can a 2nd Order In-Homogeneous Particular Solution have complex roots?

Yes, a 2nd Order In-Homogeneous Particular Solution can have complex roots. This is especially common when using the method of undetermined coefficients, as the particular solution may involve trigonometric or exponential functions.

5. How is a 2nd Order In-Homogeneous Particular Solution used in science?

A 2nd Order In-Homogeneous Particular Solution is used in science to model various physical systems, such as oscillating motion or electrical circuits. By finding the particular solution to a differential equation, scientists can accurately predict the behavior of these systems and make further observations or experiments to test their theories.