INSTRUCTOR: Verify that u of x, y, t equal to 2 sine parentheses x over 3 close parentheses sine parentheses y over 4 close parentheses e to the negative 25t over 16 end exponent is a solution to the heat equation ut equals 9 open parentheses uxx plus uyy close parentheses. In order to solve this question, we need three things, the partial of u with respect to t, the second partial of u with respect to x, and then x again, the second partial uyy, and then we can verify this equation holds. So first let us find ux. ux treating the portions of this that are not in terms of x as constants. This becomes 2/3 sine open parentheses y over 4 e to the negative 25t over 16 end exponent I'm [? told ?] by cosine open parenthesis x over 3 close parentheses, which means that uxx will be equal to negative 2/9 sine open parentheses y over 4 close parentheses e to the negative 25t over 16 multiply by sine open parenthesis x over 3.

Next, we'll find uy. And that is equal to one half sine open parenthesis x over 3 close parentheses e to the negative 25t over 16 end exponent cosine open parentheses y over 4, which means that uyy will be equal to negative 1/8. Yes, that's right. That will be equal to negative 1/8 sine open parenthesis x over 3 closed parentheses e to the negative 25t over 16 end exponent sine open parentheses why over 4 close parentheses.

Now, if we sum these two, uxx and uyy, and then multiply by 9, let's say 9 open parentheses uxx plus uyy, that will be equal to-- well, let's see. What will they be equal to? Notice these have the same components. You have a sine of y over 4, a sine of x over 3 and an e to the negative 25t over 16. So we can sum the coefficients and that would be equal to [? says ?] negative 2 over 9 minus 1 over 8 [? multiplied ?] by 9.

That is negative 25 over 8. So negative 25 over 8 sine open parenthesis x over 3 close parentheses sine open parentheses y over 4 close parentheses e to the negative 25t over 16 end exponent. Then, finally, to compare this, you need to find ut partial of u with respect to t. And that is equal to negative 25 over 8 sine open parenthesis x over 3 close parentheses sine of y over 4-- sine open parentheses y over 4 close parentheses e to the negative 25t over 16 end exponents. And since those coefficients are equal, this function, u of x, y, t is in fact a solution to this equation.