solve the heat equation ut = kuxx -infinity < x < infinity and 0 < t < infinity with u(x,0)= x2 and uxxx(x,0)= 0 first i showed that uxxx(x,t) solves the equation (easy part) the next step is to conclude that u(x,t) must be of the form A(t)x2 + B(t)x + C(t). i tried to do this by integrating uxxx(x,t) with respect to x and i got uxx(x,t) + A(t). then i solved for uxx(x,t), integrated it with respect to x, and so on until i got u(x,t) = (triple integral of uxxx(x,t) with respect to x) + A(t)x2/2 - B(t)x - C(t). It seems like the only way to get rid of the uxxx(x,t) is to say that it is identically equal to 0 since uxxx(x,0)= 0. Then i can get the desired form of the solution. But, can you say that uxxx(x,t) is identically equal to 0? Once I show that A(t)x2 + B(t)x + C(t) is the form of the equation, I'm supposed to use the initial conditions to find A(t), B(t), and C(t). But, the initial conditions will only tell me what A(0), B(0), and C(0) are.That's another problem and lastly, this may have little to do with the problem but if u(x,0)= x2 then does it mean that ux(x,0)= 2x just by differentiating both sides with respect to x or are there special conditions that must be satisfied?