Proof on Linear 1st Order IVP solution being bounded

[marvalos]

Proof on Linear 1st Order IVP solution being "bounded"

A function h(t) is called "bounded" for t≥t0 if there is a constant M>0 such that

|h(t)|≤M for all t≥0

The constant M is called a bound for h(t). Consider the IVP

x'=-x+q(t), x(0)=x0

where the nonhomogeneous term q(t) is bounded for t≥0. Show the solution of this IVP is bounded for t≥0. (Hint: Use the Variation of Constants Formula.)

Any help on where to go for this problem would be great. Thanks

 

[HallsofIvy]

You are given the "Hint: use the Variation of Constants Formula". Okay, what is that formula?

 

[marvalos]

The Variation of Constants formula is a generalized formula for First Order Linear DE's that can be solved with the Integrating Factor Method.

I would put the exact formula down but I am not too familiar with this equation editor.

 

[LCKurtz]

You have to show some effort. Show us what you have tried. I would give a second hint: It is a linear equation.

 

[marvalos]

Here is what I have tried:

The Variation of Constants formula gave me this

x=(x0+∫e^u q(u) du) e^-t

the integral is definite and goes from 0 to t.

Since q(t) is bounded, would that remain true if the integral is taken from it?

 

[LCKurtz]

Here is what I have tried:

The Variation of Constants formula gave me this

x=(x0+∫e^u q(u) du) e^-t

the integral is definite and goes from 0 to t.

Since q(t) is bounded, would that remain true if the integral is taken from it?

To rephrase your question, if you take the absolute value of both sides of that equation, can you overestimate the right hand side by some constant. So try it and show us what happens.