PDEs diffusion eq question

Hi,

I don’t really know where to go from this point, it asks for simultaneous equations of F(x) and H(x) but I’ve got derivatives of the function as well so I’m not quite sure what to do? Not sure how I am supposed to get two simultaneous equations from this

Q

I think you have a term missing on the right as the first derivative is roughly
x*e(-x^2)
so youll get two terms when differentiating again. For the two simultaneous equations, Id guess youd argue that equating the x^2 terms and the non x^2 terms as seperate (simultaneous) equations and solving, though cant quite explain properly at the moment and about to pack it in.

I think you have a term missing on the right as the first derivative is roughly
x*e(-x^2)
so youll get two terms when differentiating again. For the two simultaneous equations, Id guess youd argue that equating the x^2 terms and the non x^2 terms as seperate (simultaneous) equations and solving, though cant quite explain properly at the moment and about to pack it in.

I See but what about the x term? Because you can’t equate F(t) to the x term?

The second derivaitve on the right will get a
constant + x^2
terms (constant is in terms of x). So its similar to equating polymonimal coeffs, equate the two parts with x^2 multipliers, and equate the two "constant x" terms which looks like it will give, but not worked it through. You can do this because all the other terms are functions of t, so the x^2 terms must match as well as the cosntant terms. Altertnatively as the equation must hold for all x, it must hold with x=0 and with x=1, for instance.