Solving 2nd-Order ODEs: y'' + 2y' + y = f(t); y0=y0'=0

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In summary, the problem is to find a function y(t) with given initial conditions that satisfies the differential equation y'' + 2y' + y = f(t), where f(t) is a piecewise function. The solution involves integrating the Green's function G(t,t') with bounds 0 to infinity, using the given piecewise function as the input.
  • #1
Lanza52
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Homework Statement


y'' + 2y' + y = f(t); y0=y0'=0
f(t) is piecewise -- 1 for 0 < t < a; 0 for t > a

Use
y(t) = ∫G(t,t') f(t') dt' with bounds 0 to infinity


2. The attempt at a solution

I don't really have any logical attempt. My highest math is diffy q 1, Calc 3 and LA 1, I can't find a single resource (besides the 2 pages in my textbook) that addresses this topic in language that I understand.
 
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  • #2
Lanza52 said:

Homework Statement


y'' + 2y' + y = f(t); y0=y0'=0
f(t) is piecewise -- 1 for 0 < t < a; 0 for t > a

Use
y(t) = ∫G(t,t') f(t') dt' with bounds 0 to infinity


2. The attempt at a solution

I don't really have any logical attempt. My highest math is diffy q 1, Calc 3 and LA 1, I can't find a single resource (besides the 2 pages in my textbook) that addresses this topic in language that I understand.
See the Example section on this page - http://en.wikipedia.org/wiki/Green's_function.
 

1. What is a 2nd-order ODE?

A 2nd-order ODE, or second-order ordinary differential equation, is a mathematical equation that involves an unknown function and its derivatives up to the second order. In this case, the equation is y'' + 2y' + y = f(t).

2. What is the meaning of y0=y0'=0 in this equation?

The initial conditions y0=y0'=0 mean that the function y and its first derivative y' both have a value of 0 at the starting point t=0. This information is necessary to uniquely solve the equation.

3. How do you solve a 2nd-order ODE?

To solve a 2nd-order ODE, you need to first find the general solution, which is the most general form of the solution that satisfies the equation. This can be done through various methods such as separation of variables, substitution, or using a specific formula for the type of equation. Then, you can use the initial conditions to find the particular solution that satisfies the specific starting conditions of the equation.

4. What is the purpose of f(t) in the equation?

The function f(t) is known as the forcing function or the source term. It represents any external influences or forces acting on the system and affects the behavior of the solution y(t).

5. What are some real-life applications of 2nd-order ODEs?

2nd-order ODEs are commonly used in physics and engineering to model the motion of objects under the influence of forces, such as oscillations in a pendulum or vibrations in a spring. They are also used in electrical circuits, population growth models, and many other areas of science and engineering.

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