Verifying Solution of PDE utt = c2uxx with FTC

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In summary, the given function u(x,t) is a solution to the PDE utt = c2uxx by using the Fundamental Theorem of Calculus instead of the Leibniz Rule to take derivatives. This involves taking the derivative of the integral and using the fundamental theorem to evaluate it.
  • #1

Homework Statement



Verify that, for any continuously differentiable function g and any constant c, the function

u(x, t) = 1/(2c)∫(x + ct)(x - ct) g(z) dz ( the upper limit (x + ct) and lower limit (x - ct))

is a solution to the PDE utt = c2uxx.

Do not use the Leibnitz Rule, but instead review the

Fundamental Theorem of Calculus.

Homework Equations


Fundamental Theorem of Calculus I & II.


The Attempt at a Solution


Not a clue but tried the first and second fundamental theorem of calculus (learned in calc I or II) but did not seem to get anywhere.
 
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  • #2
proximaankit said:

Homework Statement



Verify that, for any continuously differentiable function g and any constant c, the function

u(x, t) = 1/(2c)∫(x + ct)(x - ct) g(z) dz ( the upper limit (x + ct) and lower limit (x - ct))

is a solution to the PDE utt = c2uxx.

Do not use the Leibnitz Rule, but instead review the

Fundamental Theorem of Calculus.

Homework Equations


Fundamental Theorem of Calculus I & II.


The Attempt at a Solution


Not a clue but tried the first and second fundamental theorem of calculus (learned in calc I or II) but did not seem to get anywhere.

This question is simply about taking derivatives and solving the equation. If you can't recall the fundamental theorem, here's a nutshell version of it :

$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(t) dt = f(b(x))(b'(x)) - f(a(x))(a'(x))$$
 

1. What is the FTC?

The FTC, or Fundamental Theorem of Calculus, is a fundamental principle in calculus that relates the concept of differentiation with that of integration. It states that the integral of a function can be evaluated by finding the antiderivative of the function and evaluating it at the endpoints of the interval.

2. Why is the FTC important in verifying solutions of PDE utt = c2uxx?

The FTC is important in verifying solutions of PDE utt = c2uxx because it allows us to evaluate the integral of the solution function and compare it to the derivative of the function. This helps us verify if the solution satisfies the given PDE.

3. How does the FTC help in verifying solutions of PDE utt = c2uxx?

The FTC helps in verifying solutions of PDE utt = c2uxx by providing a mathematical framework for evaluating the integral of the solution function and comparing it to the derivative of the function. If the two are equal, then we can say that the solution satisfies the PDE.

4. Can the FTC be applied to all PDEs?

No, the FTC can only be applied to PDEs that can be written in terms of a single independent variable. PDEs with multiple independent variables require different methods for verifying solutions.

5. How does the FTC relate to solving PDE utt = c2uxx?

The FTC does not directly relate to solving PDE utt = c2uxx, but it can be used as a tool to verify the solutions obtained through other methods. It is an important concept in calculus and can help in understanding the behavior of solutions to PDEs.

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