2nd Order Linear Diff. Eqn (homogeneous)

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?

Answers and Replies

Defennder
Homework Helper
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?

Defennder
Homework Helper
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
Science Advisor
Homework Helper
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)?