Laplace transform and bessels equation

• wtmoore
In summary, the conversation discusses finding the Laplace transform of tJ''0(t), which involves integrating (e^-st)*t*J''0(t). Maple 14 provides a solution by using the second derivative of J0(t), times t, and expanding it. The resulting integral is then expressed as a function of s and solved using Maple's differential equations and recursions.
wtmoore

Homework Statement

I'm trying to find the Laplace transform of tJ''0(t), it's from bessels equation, but that doesn't matter too much at the moment, I just need to integrate (e^-st)*t*J''0(t) but am unsure how to go about this with the J''0(t) in there.

t*diff(BesselJ(0,t),t\$2): <---second derivative of J0(t), times t

f:=expand(%);
f := -t BesselJ(0, t) + BesselJ(1, t)

L:=int(f*exp(-s*t),t=0..infinity);

L:= [1 + 2s^2 + s^4 - (s^2 + 2s)*sqrt(1 + s^2)]/ (1+s^2)^2

Note: Maple used the DE for J0 and recursions to express J0'' in terms of J0 and J1, then it integrated that.

RGV

1. What is the Laplace transform?

The Laplace transform is a mathematical tool used to convert a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems.

2. How is the Laplace transform calculated?

The Laplace transform is calculated by integrating a function of time multiplied by a complex exponential e^(-st), where s is a complex number. This results in a new function of complex frequency s.

3. What are the applications of Laplace transform?

Laplace transform has many applications in engineering, physics, and mathematics. It is commonly used to solve differential equations, analyze control systems, and study signals and systems in communication and signal processing. It is also used in probability and statistics to find the moment-generating function of a random variable.

4. What is Bessel's equation?

Bessel's equation is a second-order ordinary differential equation that arises in many problems involving cylindrical or spherical symmetry. It is named after the mathematician Friedrich Bessel and has important applications in physics and engineering, such as in the study of heat conduction and vibration of circular membranes.

5. How is Bessel's equation related to the Laplace transform?

Bessel's equation is related to the Laplace transform through the Bessel functions, which are the solutions to Bessel's equation. These functions are used to express the solutions to many physical problems involving cylindrical or spherical symmetry, and they can be obtained from the Laplace transform of certain functions.

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