PDE homework

  • #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.
 

Answers and Replies

  • #2
STEMucator
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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 :

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

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