_ represents a subtext
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For this to be the case for all x,t we would need u_t and u_xx to be a constant,
so we would require u(x,t) = k*t + (k/2)*x^2 + C where k and C are constants.
To go any further would require some initial and boundary conditions.
Edit: Hmm, sorry, there are more possible solutions. Look at a solution of the
form u(x,t) = e^(ax + bt). Then for u_t to equal u_xx we would require that
b*u = a^2*u ---> b = a^2. Thus u(x,t) = e^(ax + (a^2)*t) for any a would also
be a solution.
u_t = u_xx is known as the heat equation, and has many different solutions
depending upon the initial and boundary conditions.
so we would require u(x,t) = k*t + (k/2)*x^2 + C where k and C are constants.
To go any further would require some initial and boundary conditions.
Edit: Hmm, sorry, there are more possible solutions. Look at a solution of the
form u(x,t) = e^(ax + bt). Then for u_t to equal u_xx we would require that
b*u = a^2*u ---> b = a^2. Thus u(x,t) = e^(ax + (a^2)*t) for any a would also
be a solution.
u_t = u_xx is known as the heat equation, and has many different solutions
depending upon the initial and boundary conditions.
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I really don't know