2nd Order Linear Diff. Eqn (homogeneous)

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Homework Help Overview

The problem involves a second-order linear differential equation of the form y'' + p(t)y' + q(t)y = g(t), where g(t) is not always zero, indicating that the equation is non-homogeneous. The original poster seeks to demonstrate that if y = x(t) is a solution, then y = c*x(t) (for any constant c other than 1) cannot be a solution.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss substituting y = c*x(t) into the differential equation to analyze the implications. Questions arise about the behavior of the equation under this substitution and the role of the constant c.

Discussion Status

The discussion is ongoing, with participants exploring the consequences of substituting y = c*x(t) into the differential equation. There is a focus on understanding how the presence of the constant affects the equality and the implications for the solution.

Contextual Notes

Participants note that since g(t) is not zero, the equation is confirmed to be non-homogeneous, which is a critical aspect of the problem being discussed.

aznkid310
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Homework Statement


Show that id y = x(t) is a solution of the diff. eqn. y'' + p(t)y' + q(t)y = g(t), where g(t) is not always zero, then y = c*x(t), where c is any constant other than 1, is not a solution.


Homework Equations


Can someone help me get started?
Also, since g(t) is not zero, this means that the equation is nonhomogeneous?


The Attempt at a Solution

 
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Yes, this implies the DE is non-homogenous. To show that y = c*x(t) is not a solution just substitute that into the DE. Do you still get g(t) on the RHS?
 
For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?
 
Note that it is given that y=x(t) is a solution, that means x''(t) + p(t)x'(t) + q(t)x(t) = g(t). The expression you get when you substitute y = cx(t) into the DE is clearly different from this. What does that tell you?
 
aznkid310 said:
For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?
What happened to the c?? If x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t) what is
(c*x)'' + p(t)*(c*x)' + q(t)*(c*x)?
 

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