I need help solving linear wave functions?

Question

part (a): y(x,t) = x^2 + (v^2)(t^2)

Show that the function in part (a) can be written as f(x + vt) + g(x - vt), and determine the functional forms for f and g

f(x+vt)=?

g(x-vt)=?

my understanding is that :

f(x+vt)+g(x-vt)= x^2+(v^2)(t^2) and that is about as far as i get.

Answer 1

y = x^2 +(vt)^2

y = f(x+vt) + g(x-vt)

let f = A(x+vt)^2 and g = B(x-vt)^2

(A+B) x^2  + 2(A-B)xvt +(A+B) (vt)^2 = x^2 + (vt)^2

Equate power of x, vt on both sides:

A+B = 1  for x^2 and for (vt)^2 terms

A - B = 0 for xvt term  ---> A = B

So 2A = 1 ---> A= 1/2 = B

thus

x^2+(vt)^2 = 1/2{(x+vt)^2 +(x-vt)^2}

Answer 2

Well, you can define:

a = x + vt

b = x - vt

From here it is obvious that:

x = (a + b) / 2

vt = (a - b) / 2

And explain y(x,t) in terms of a and b:

y(x,t) = ((a+b)/2)^2 + ((a-b)/2)^2 = a^2 / 2 + b^2 / 2

So:

f(x+vt) = (x+vt)^2 / 2

g(x-vt) = (x-vt)^2 / 2

y(x,t) = ((x+vt)^2 + (x-vt)^2) / 2