2nd Order Linear Diff. Eqn (homogeneous)

  • #1
109
1

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

 
  • #2
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?
 
  • #3
For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?
 
  • #4
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?
 
  • #5
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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