Question

using chain rule, verify any function of the form z(u,v)= f(u) + g(v)
where
u(x,t)=x+at , v(x,t)=x-at
satisfied the heat equation (partial differential equation) given by
\\[z _{t t}=a ^{2}z _{xx}\]

Answer 1

@skullpatrol @rvc @roadjester @gyanu @Hero @theEric

Answer 2

Please help someone :s i am stuck with this..

Answer 3

@ganeshie8 @ash2326 @sarah786

Answer 4

not a good maths Helper :/

Answer 5

@sarah786 then please direct someone here who you know :)
Regards

Answer 6

@esshotwired @ganeshie8 @kc_kennylau

Answer 7

:( someone please help me

Answer 8

Well, let's do what the problem says to do, and apply the chain rule. In particular, we get that\[\frac{d}{dt}z(u,v)=\frac{dz}{du}\frac{du}{dt}+\frac{dz}{dv}\frac{dv}{dt}\]Plugging in the respective values, this becomes\[\frac{d}{dt}z(u,v)=f'(u)a+g'(v)a.\]Repeat the process once more to find \(z_{tt}\), and do the whole thing over for \(z_{xx}\).

Answer 9

oh thank you so much @KingGeorge i will get back after completing what you told me :)
thanks again

Answer 10

@KingGeorge Will you please tell me, if i am doing this right
f'(u)=a and g'(v)=-a
but
f''(u) will become 0 and so will g''(v) :s

Answer 11

You don't want to say that \(f'(u)=a\). We don't know what the function \(f\) is. We only know that it's a function of \(u\), and \(u\) itself is a function of \(x\) and \(t\). Just leave\(f'(u)\) as it's written.

Answer 12

alright, but when we go for \[z _{t t }\] we will get
a f''(u) + a g''(v)?

Answer 13

It shouldn't be exactly that, but it'll be close to that.

Answer 14

okay and what about
\[z_{ x x }\]
will it be close to
f''(u) + g''(v)?

then how will we get
\[z_{ t t }=a^2 * z_{ x x }\]?

thanks again for quick response and sorry for being so bad at it

Answer 15

For \(z_{xx}\), you should get exactly \(f''(u)+g''(v)\), and for \(z_{tt}\), it should be \(a^2f''(u)+a^2g''(v)\). (You should check this yourself). From then, it's just a simple matter of noticing\[a^2z_{xx}=a^2f''(u)+a^2g''(v)=x_{tt}.\]

Also, I made a slight typo above. We should have\[z_t=af'(u)-ag'(v).\]Notice the minus sign.

Answer 16

Thank you very much @KingGeorge
<3

Answer 17

You're welcome.