Is F(x,t) = f(x-ct) , where f is any function with at least 2 derivatives,?

Question

a solution to the partial differential equation:

c^2 * d^2 u / dx^2  -  d^2 u / dt^2 = 0.

Justify your answer. You will need to use chain rule.

Thanks!

schmiso · Accepted Answer

Let
z = x - c∙t
Then
u = F(x,t) = f(z)
=>
∂u/dx = (∂u/∂z)∙(∂z/dx) = (df/dz)∙(∂z/dx) = (df/dz)∙1
Note that df/dz is not a partial differential because f is function of z alone)
∂²u/∂x² = ∂( df/dz ) )//∂z)∙(∂z/dx) = (d²f/dz²)∙1 = (d²f/dz²)
and
∂u/dt = (∂u/∂z)∙(∂z/dt) = (df/dz)∙(∂z/dx) = (df/dz)∙c
∂²u/∂t² = ∂( c∙(df/dz) ) )//∂z)∙(∂z/dx) = c∙(d²f/dz²)∙c = c²∙(d²f/dz²)

Hence,
c²∙(∂²u/∂x²) - (∂²u/∂t²) 
= c²∙(d²f/∂z²) - c²∙(d²f/dz²) 
= 0
q.e.d.

That means f(x - c∙t) is a solution to this partial differential equation.

Trending News

Is F(x,t) = f(x-ct) , where f is any function with at least 2 derivatives,?

1 Answer