Verifying Solution of PDE utt = c2uxx with FTC

[proximaankit]

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 u_tt = c²u_xx.

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.

 

[STEMucator]

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 u_tt = c²u_xx.

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 :

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