Help solving the heat equation

[timjones007]

help solving the heat equation!

solve the heat equation
u_t = ku_xx

-infinity < x < infinity and 0 < t < infinity

with u(x,0)= x² and u_xxx(x,0)= 0

first i showed that u_xxx(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 u_xxx(x,t) with respect to x and i got u_xx(x,t) + A(t). then i solved for u_xx(x,t), integrated it with respect to x, and so on until i got u(x,t) = (triple integral of u_xxx(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 u_xxx(x,t) is to say that it is identically equal to 0 since u_xxx(x,0)= 0. Then i can get the desired form of the solution. But, can you say that u_xxx(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 u_x(x,0)= 2x just by differentiating both sides with respect to x or are there special conditions that must be satisfied?

 

[gtoguy97]

Yes, you can say that uxxx(x,t) is identically equal to 0 if uxxx(x,0)= 0. For the initial conditions, you can use them to determine the coefficients A(0), B(0), and C(0). Then, you can use the heat equation to solve for A(t), B(t), and C(t).Yes, since u(x,0)= x2, then ux(x,0)= 2x.