Suppose f is a function of one variable that has a continuous second derivative. Show that for any constants a and b, the function u(x, y) = f(ax + by) is a solution of the PDE uxxuyy −u2xy = 0.

We can find the solution using chain rule,

For instance, if u(x,y) = f(ax+by), then

 which represents derivative of x with respect to x

and then derivative of with respect to y is

,  

Now, the derivative of with respect to x, which is the second derivative, which is

and the derivative and   are

,

Finally, the solution of PDE is

= 0,

As the PDE is equal to 0, it means the function u(x,y) = f(ax+by) is the solution of the given PDE.

constants mean the stopped points or the points on a graph that nothing is changing, hope this

step-by-step explanation: