What is the solution to a nonhomogenous second order differential equation?

In summary, the conversation discusses solving a nonhomogeneous second order differential equation with the equation y'' - 2y' - 3y = -3te^{-t}. Different attempts at solutions are made, such as Y(t) = Ate^{-t}, Y(t) = At^2e^{-t}, and Y(t) = Ae^{-t}. Eventually, the correct solution is found to be At^2e^{-1} + Bte^{-1}.
  • #1
jesuslovesu
198
0
[SOLVED] Nonhomogenous Second Order DE

Homework Statement


[tex]y'' - 2y' - 3y = -3te^{-t}[/tex]
Solve

Homework Equations


The Attempt at a Solution



Well first I tried
[tex]Y(t) = Ate^{-t}[/tex]
then
[tex]Y(t) = At^2e^{-t}[/tex]
then
[tex]Y(t) = At^3e^{-t}[/tex]
and
[tex]Y(t) = At^3e^{-t} + Bt^2e^{-t}[/tex]
but I can't seem to get the correct form.

Does anyone know what form I need my Y(t) in order to solve this?
 
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  • #2
I believe I end up with a term like
[tex]Ae^{-t} = -3te^{-t}[/tex] which I don't believe I can do anything with?

In fact, in all the cases I've tried I always seem to get something like [tex]Ae^{-t}[/tex] which never cancels out so I can never equation -3te^{-t} to any of my terms
 
Last edited:
  • #3
ah finally got it with [tex]At^2e^{-1}+Bte^{-1}[/tex]
bah that sucked
 

FAQ: What is the solution to a nonhomogenous second order differential equation?

What is a nonhomogeneous second order differential equation?

A nonhomogeneous second order differential equation is a mathematical equation that involves a second derivative of a function and contains terms that are not proportional to the function itself or its derivatives. These terms are referred to as nonhomogeneous terms or forcing terms. Unlike homogeneous equations, which have a solution that is a linear combination of exponential functions, nonhomogeneous equations require additional techniques to solve.

How do you solve a nonhomogeneous second order differential equation?

To solve a nonhomogeneous second order differential equation, one method is to first solve the corresponding homogeneous equation. Next, a particular solution is found for the nonhomogeneous equation using methods such as variation of parameters, undetermined coefficients, or the method of annihilators. Finally, the general solution is obtained by combining the particular solution with the solution to the homogeneous equation.

What are the applications of nonhomogeneous second order differential equations?

Nonhomogeneous second order differential equations are used to model a variety of physical phenomena in fields such as physics, engineering, and economics. Some common applications include the motion of a spring-mass system, electrical circuits, and population growth models. These equations can also be used to predict the behavior of systems over time and make important decisions based on the solutions.

What are the boundary conditions for solving nonhomogeneous second order differential equations?

The boundary conditions for solving nonhomogeneous second order differential equations depend on the specific problem being modeled. Generally, there are two types of boundary conditions: initial conditions and boundary value conditions. Initial conditions specify the values of the function and its derivatives at a single point, while boundary value conditions specify the values of the function and its derivatives at multiple points. These conditions are necessary to obtain a unique solution to the differential equation.

How do nonhomogeneous second order differential equations differ from first order equations?

Nonhomogeneous second order differential equations are more complex than first order equations because they involve a second derivative of the function. This means that the solution to the equation will have two arbitrary constants, compared to only one for first order equations. Additionally, the methods used to solve nonhomogeneous second order equations are more involved and require knowledge of linear algebra and the theory of differential equations.

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