2nd Order Linear Diff. Eqn (homogeneous)

[aznkid310]

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

 

[Defennder]

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?

 

[aznkid310]

For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?

 

[Defennder]

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?

 

[HallsofIvy]

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)?