hello,
im having difficulty answering this question, please could somebody spare sometime answering it.. it starts off an exam paper and is putting me off the whole thing..
wats da general solution of
(∂^2/∂x∂t )f(x,t)=xt
thanks in advance
hello,
im having difficulty answering this question, please could somebody spare sometime answering it.. it starts off an exam paper and is putting me off the whole thing..
wats da general solution of
(∂^2/∂x∂t )f(x,t)=xt
thanks in advance
Do you mean $\displaystyle \frac{\delta^2 f}{\delta x \delta t} = xt $? In which case, integrating once with respect to t,
$\displaystyle \frac{\delta f}{\delta x} = \frac{xt^2}{2} + A'(x) $ where A is arbitrary. Integrating again with respect to x we get
$\displaystyle f(x,t) = \frac{(xt)^2}{4} + A(x) + B(t) $ With B also being arbitrary.
thank-you for ur reply.. not too sure why you have got A'(x) in the first part.. but then for the B arbitary constant you have B(t), shouldnt this be B'(t) too. I underdstand why A'(x) becomes A(x), or maybe i have not got the concept wright????
thanks in advance
Okay, so, when we integrate with respect to t, we have to include an arbitrary function of x. You can call this function A or A', because it's completely arbitrary what it is anyway. We choose A' because we know that we still have to integrate with respect to x, which will yield just A.
After integrating with respect to x, we need to include an arbitrary function of t. We choose B(t) instead of B'(t) because we aren't going to be integrating again. You could have chosen to call it B' and you'd still be correct - it's an arbitrary function anyway. I hope that clears up my choices here.