Fun Little Differential Equation Exercise

Saw this problem in Zill, 6th Edition (it's Problem 4.3.62, in case you're curious), and had enough fun solving it that I thought I'd post it here for your pleasure. It's not an overly challenging problem, but it is more subtle than it appears at first.

What conditions should be imposed on the constant coefficients and in order to guarantee that all solutions of the second-order differential equation are bounded on the interval (Assume the coefficients are all real.)

Re: Fun Little Differential Equation Exercise

Quote:

Originally Posted by

**Ackbeet** Saw this problem in Zill, 6th Edition (it's Problem 4.3.62, in case you're curious), and had enough fun solving it that I thought I'd post it here for your pleasure. It's not an overly challenging problem, but it is more subtle than it appears at first.

What conditions should be imposed on the constant coefficients

and

in order to guarantee that all solutions of the second-order differential equation

are bounded on the interval

(Assume the coefficients are all real.)

The condition is that both the solutions of the algebraic equation do have real part not greater than 0. The same is for a non-homogeneous linear DE with constant coefficient of any order. For an homogeneous requation like , where h(*) is a bounded function in the condition is that both the solutions of do have real part *strictly* less than 0...

Kind regards

Re: Fun Little Differential Equation Exercise

Quote:

Originally Posted by

**chisigma** The condition is that both the solutions of the algebraic equation

do have real part not greater than 0. The same is for a non-homogeneous linear DE with constant coefficient of any order. For an homogeneous requation like

, where h(*) is a bounded function in

the condition is that both the solutions of

do have real part

*strictly* less than 0...

Kind regards

All very true. However, the question is asking for conditions on a, b, and c; not on the characteristic equation. So I think the question is asking you to go further.

Re: Fun Little Differential Equation Exercise

assume a>0

if

then b>0

if

then

Re: Fun Little Differential Equation Exercise

Quote:

Originally Posted by

**wnvl**

What about ? Can that happen and the solution still be bounded?

Quote:

assume a>0

if

then b>0

if

then

What if What if Can you then generalize a condition that works for every subcase?

Re: Fun Little Differential Equation Exercise

Quote:

Originally Posted by

**Ackbeet** What about

? Can that happen and the solution still be bounded?

if then a bounded solution is only possible if .

Quote:

Originally Posted by

**Ackbeet** What if

What if

Can you then generalize a condition that works for every subcase?

If a = 0, then we need -c/b < 0 for bounded solution.

If i then we need b>0.

Re: Fun Little Differential Equation Exercise

Quote:

Originally Posted by

**chisigma** The condition is that both the solutions of the algebraic equation

do have real part not greater than 0. The same is for a non-homogeneous linear DE with constant coefficient of any order. For an homogeneous requation like

, where h(*) is a bounded function in

the condition is that both the solutions of

do have real part

*strictly* less than 0...

Kind regards

A polynomial the roots of which have all non-positive real part is called *Hurwitz Polynomial*. Neccesary but not sufficient condition for a polynomial to be Hurvitz is that all coefficients have the same sign. For neccesary and suffiicient conditions see...

Hurwitz polynomial

Routh Hurwitz stability criterion

Kind regards

Re: Fun Little Differential Equation Exercise