2nd Order ODE Initial Value Proof Problem

In summary, the given function y(t) satisfies the initial value problem y''(t)+λ^2y(t)=f(t) for λ>0 and y(0)=0, y'(0)=0. Though the solution involves an integral, it does not need to be explicitly integrated. Instead, the function can be differentiated and plugged into the differential equation to prove its validity. The function can also be seen as the convolution of the Green's function of the undamped oscillator with the inhomogeneity on the right-hand side. Differentiating a definite integral can be done using the method of differentiation under the integral sign.
  • #1
DuncP89
4
0
1. Check that

y(t)=1/λ ∫_0-t_〖f(s) *sin(λ(t-s) )ds〗
is the solution of the following initial value problem

y''(t)+λ^2y(t)=f(t), λ>0, y(0)=0,y'(0)=0


Homework Equations





3. I tried to do integration by parts on y(t), but just doesn't work. I'm not sure how to prove it using those two initial conditions
 
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  • #2
Try using Laplace transforms.
 
  • #3
SteamKing said:
Try using Laplace transforms.

I have never learned laplace transformation, I'm sure my prof is not looking for anything like that
 
  • #4
You are not asked to integrate or invent Laplace transforms, only to differentiate something, with the advantage of being told the answer.
 
Last edited:
  • #5
epenguin said:
You are not asked to integrate or invent Laplace transforms, only to differentiate something, with the advantage of being told the answer.

Forgive me, I don't quite understand it. How is it only about differentiate? There is an integral on the answer y(t), you can get rid of it on y''(t), but I don't know how to get rid of it on λ^2*y(t)
 
  • #6
Obviously, you can't. All you can do is multiply λ^2 by y(t).
 
  • #7
I haven't tried it myself, but if I understand the question correctly, you should not try to do the integral but just take the derivatives wrt. [itex]t[/itex] and prove that it fulfills the differential equation and the initial conditions.

On the other hand, that's pretty clear since, what's written there is nothing than the convolution of the Green's function of the undamped oscillator-differential operator with the inhomogenity on the right-hand side, and thus the claim in the problem is correct.
 
  • #8

1. What is a 2nd Order ODE Initial Value Proof Problem?

A 2nd Order ODE Initial Value Proof Problem is a mathematical problem that involves solving a second-order ordinary differential equation (ODE) with initial conditions. The goal is to find the solution that satisfies both the differential equation and the given initial conditions.

2. What is the difference between a 1st and 2nd Order ODE Initial Value Proof Problem?

The main difference between a 1st and 2nd Order ODE Initial Value Proof Problem is the order of the differential equation. A first-order ODE has only one derivative, while a second-order ODE has two derivatives. This means that a 2nd Order ODE Initial Value Proof Problem requires more information (two initial conditions) to find a unique solution compared to a 1st Order ODE Initial Value Proof Problem (only one initial condition).

3. How do I solve a 2nd Order ODE Initial Value Proof Problem?

There are various methods for solving a 2nd Order ODE Initial Value Proof Problem, including the method of undetermined coefficients, variation of parameters, and Laplace transforms. The specific method used depends on the form of the differential equation and the given initial conditions.

4. Why are initial conditions necessary for solving a 2nd Order ODE Initial Value Proof Problem?

Initial conditions provide the necessary information to find a unique solution to the 2nd Order ODE Initial Value Proof Problem. Without initial conditions, the solution could have an infinite number of possibilities, making it impossible to determine the exact solution.

5. What are some real-life applications of 2nd Order ODE Initial Value Proof Problems?

2nd Order ODE Initial Value Proof Problems have applications in various fields, including physics, engineering, and economics. Some examples include modeling the motion of a pendulum, analyzing the vibrations of a guitar string, and predicting the growth of a population over time.

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