What kind of differential equation is this?

[mathman44]

Homework Statement

It's been a long time since I've done DE's and now I'm stuck with a monster of this form:

y'(t) = a*g'(t) + b*g(t) + c*y(t)

where g(t) is a known function and a, b and c are constants. What kind of DE is this, and how can I solve for y(t) -- or better yet, what should I read to figure this out? The answer involves a convolution integral, that I know for sure.

Cheers.

 

[Ray Vickson]

Homework Statement

It's been a long time since I've done DE's and now I'm stuck with a monster of this form:

y'(t) = a*g'(t) + b*g(t) + c*y(t)

where g(t) is a known function and a, b and c are constants. What kind of DE is this, and how can I solve for y(t) -- or better yet, what should I read to figure this out? The answer involves a convolution integral, that I know for sure.

Cheers.

This is just a simple linear DE of the form y'(t) - c*y(t) = f(t), where f(t) is the KNOWN function f(t) = a*g'(t) + b*g(t). There is nothing at all monstrous about it.

 

[Dick]

Homework Statement

It's been a long time since I've done DE's and now I'm stuck with a monster of this form:

y'(t) = a*g'(t) + b*g(t) + c*y(t)

where g(t) is a known function and a, b and c are constants. What kind of DE is this, and how can I solve for y(t) -- or better yet, what should I read to figure this out? The answer involves a convolution integral, that I know for sure.

Cheers.

If g(t) is known, then if you put h(t)=a*g'(t)+b*g(t), it's y'(t)-c*y(t)=h(t). It's an inhomogeneous first order linear equation. Nothing to do with convolution I can see.

 

[Ray Vickson]

If g(t) is known, then if you put h(t)=a*g'(t)+b*g(t), it's y'(t)-c*y(t)=h(t). It's an inhomogeneous first order linear equation. Nothing to do with convolution I can see.

Well, the formula for the nonhomgeneous solution *does* involve something very like a convolution. In fact, if you do it by Laplace transforms you can see explicitly that part of the solution involves a product of Laplace transforms, hence involves convolution. However, the general solution may also involve some non-convolution parts, depending on initial conditions, etc.

 

[Dick]

Well, the formula for the nonhomgeneous solution *does* involve something very like a convolution. In fact, if you do it by Laplace transforms you can see explicitly that part of the solution involves a product of Laplace transforms, hence involves convolution. However, the general solution may also involve some non-convolution parts, depending on initial conditions, etc.

Ok, shows I don't know much about Laplace transform methods. I stand corrected.

 

[Ray Vickson]

Ok, shows I don't know much about Laplace transform methods. I stand corrected.

You don't need Laplace transforms; that is just one of the ways to see it. The general solution of
$$\frac{dx(t)}{dt} - a x(t) = f(t)$$
is
$$x(t) = C e^{at} + \int_0^t e^{a(t-\tau)} f(\tau) \, d \tau.$$
Here, the first term is the solution of the homogeneous equation and the second term is a 'particular' solution to the non-homogeneous equation. The second terms is a convolution of the two functions $e^t$ and $f(t)$.