About Partial Differential equation

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[HallsofIvy]

First, do you know what the "assumptions of the Cauchy-Kovaleski theorem" are?

"A Taylor series of order less than or equal to 2 about the origin", in two variables, x and t, is $\displaystyle u(x, t)=Ax^2+ Bt^2+ Cxt+ Dx+ Et+ F$ for some contants, A, B, C, D, E, and F. What are $\displaystyle u_x$ and $\displaystyle u_t$. Put those into the equation.