Verify that the function u(x, y, z) = log x^2 + y^2 is a solution of the two dimensional laplace equation u_xx + u_yy = 0 everywhere, except of course at the origin where f is not defined. verify that the following functions solve the wave equation, u_tt = u_xx u(x, t) = cos(4x) cos(4t) u(x, t) = f(x - t) + f(x + 1), where f is any differentiable function of one variable.

The function   is indeed a solution of the two dimensional Laplace equation  .

The wave equation   is satisfied by the function but not by the function .

Step-by-step explanation:

To verify that the function   is a solution of the 2D Laplace equation we calculate the second partial derivative with respect to x and then with respect to t.

then we introduce it in the equation  

we get that  

To see if the functions 1) and 2)     solve the wave equation we have to calculate the second partial derivative with respect to x and the with respect to t for each function. Then we see if they are equal.

1)  

we see for the above expressions that  

2) with this function we will have to use the chain rule

 If we call   and    then we have that

So  

because we have   and  

then  

⇒

⇒

⇒

Regarding the derivatives with respect to time

then  

we see that  

 doesn´t satisfy the wave equation.