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u_{t}= ku_{x}_{x}

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

with u(x,0)= x^{2}and u_{x}_{x}_{x}(x,0)= 0

first i showed that u_{x}_{x}_{x}(x,t) solves the equation (easy part)

the next step is to conclude that u(x,t) must be of the form A(t)x^{2}+ B(t)x + C(t).

i tried to do this by integrating u_{x}_{x}_{x}(x,t) with respect to x and i got u_{x}_{x}(x,t) + A(t). then i solved for u_{x}_{x}(x,t), integrated it with respect to x, and so on until i got u(x,t) = (triple integral of u_{x}_{x}_{x}(x,t) with respect to x) + A(t)x^{2}/2 - B(t)x - C(t).

It seems like the only way to get rid of the u_{x}_{x}_{x}(x,t) is to say that it is identically equal to 0 since u_{x}_{x}_{x}(x,0)= 0. Then i can get the desired form of the solution. But, can you say that u_{x}_{x}_{x}(x,t) is identically equal to 0?

Once I show that A(t)x^{2}+ 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)= x^{2}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?

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# Heat equation: partial differential equations

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