Using implicit differentiation, verify that u=f(x−tu) satisfies (del u(x,t))/(del t)+u(x,t)(del u(x,t))/(del x). Could someone explain how implicit differentiation works with a pde?

madeeha1d8 2022-09-26 Answered
Using implicit differentiation, verify that u = f ( x t u ) satisfies u ( x , t ) t + u ( x , t ) u ( x , t ) x .
Could someone explain how implicit differentiation works with a pde?
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Answers (1)

geoforoiunpwd
Answered 2022-09-27 Author has 6 answers
Differentiating u = f ( x t u ) first with respect to t and then with respect to x we get:
u t = f ( x t u ) ( u t u t ) ,
u x = f ( x t u ) ( 1 t u x ) .
Multiplying the second equation by u and adding both equations we get
u t + u u x = t f ( x t u ) ( u t + u u x ) ( 1 + t f ) ( u t + u u x ) = 0.
If f 0 (i.e. f is increasing) then 1 + t f > 1 for all t 0 and u t + u u x = 0 follows.
If f < 0 at some point, there is a T max such that 1 + t f > 0 on [ 0 , T max ),
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