Question

How do I verify that each given function is a solution of the given partial differential equation? a^2u_(xx)=u_t; u=(pi/t)^(1/2)e^(-x^2/4a^2t), t>0

Answer 1

I know the function is kinda hard to see

Answer 2

do you know hot to find u_x?

Answer 3

\[u=(\frac{\pi}{t})^\frac{1}{2}e^{\frac{-x^2}{5a^2t}}\]

Answer 4

No I don't dont really understand what u_xx is

Answer 5

\[u=(\frac{\pi}{t})^\frac{1}{2}e^{\frac{-x^2}{4a^2t}}\]
\[u_x=(\frac{\pi}{t})^\frac{1}{2} (\frac{-2x}{4a^2t})e^\frac{-x^2}{4a^2t}\]

Answer 6

so i have found the partial derivative of u with respect to x

Answer 7

now i just find the partial derivative of u_x with respect to x

Answer 8

How do you find the partial derivative again?

Answer 9

\[u_{xx}=(\frac{\pi}{t})^\frac{1}{2}\frac{-2}{4a^2t}e^{\frac{-x^2}{4a^2t}}+(\frac{\pi}{t})^\frac{1}{2}\frac{-2x}{4a^2t}(\frac{-2x}{4a^2t})e^{\frac{-x^2}{4a^2t}}\]

Answer 10

you treat everything else like a constant except what you are taking the partial derivative with respect to

Answer 11

so for the first one you just look at x and u since its u with respect to x?

Answer 12

ok so now i'm going to rewrite u alittle so it is easier for me to take derivative of t (remember we are going to treat everything else like a constant except t since we are doing u_t)
\[u=\pi^\frac{1}{2}t^{-\frac{1}{2}}e^\frac{-x^2t^{-1}}{4a^2}\]
....

Answer 13

so wouldn't it be like taking the derivative of e^(1/t)?

Answer 14

\[u_t=\pi^\frac{1}{2}\frac{-1}{2}t^{\frac{-3}{2}}e^\frac{-x^2t^{-1}}{4a^2}+\pi^\frac{1}{2}t^\frac{-1}{2}\frac{x^2t^{-2}}{4a^2}e^\frac{-x^2t^{-1}}{4a^2}\]

Answer 15

Okay I'll try it first it does seem to make sense. Thanks for your help

Answer 16

I'm confuse for the first one because wouldn't you need to do the product rule? because it's like (c/t)^(1/2) *(e^(c^-1))

Answer 17

if you treat everything but t as a constant

Answer 18

i used the product rule when i was doing the partial to u respect to t
i also used the product rule when i was doing the partial to u_x with respect to x

Answer 19

but for u with respect to t don't you get ____ + _____? like two terms?

Answer 20

yes just like i did above

Answer 21

?? I only see one term for u_x

Answer 22

sorry if I'm confusing you

Answer 23

you said the parital to t?

Answer 24

so you are looking at the parital to x?

Answer 25

never mind I just did the derivative for the first one wrong, did you use like substitution? I haven't done it in awhile

Answer 26

yes once you find u_xx and u_t
then plug into
a^2u_xx=u_t

Answer 27

Okay I'm just really confused with the partial derivative of u with respect to x

Answer 28

this is everything i did:

Answer 29

you got the derivative of u considering only t and everything is a constant right

Answer 30

how do you find the derivative of
\[f(x)=5e^{-\frac{x^2}{3}}\]
?

Answer 31

u_x means you are treating everything else like a constant except x

Answer 32

Oh I thought you meant t earlier

Answer 33

\[f'(x)=5(\frac{-2x}{3})e^{\frac{-x^2}{3}}\]

Answer 34

yes we found u_x and u_t earlier

Answer 35

Oh I get it so for u_t you only look at t and for u_x you look at x

Answer 36

yes for u_t you treat everything like it is a constant except t

Answer 37

ohhh okay I get it thanks a lot for your help!! really appreciate it

Answer 38

and for u_x you treat everything like a constant except x