Completing partial derivatives to make them converge For a function f(x,y) of two independent variables we have an incomplete specification of its partial derivatives as follows: (del f(x,y))/(del x) = 1/(g(x,y) sqrt (1 - ((k y)/(x^((1/3))))^2)) (del f(x,y))/(del y) = ((3 x)/(4)) (k)/(x^((1/3)))^2 (2 y) \frac {1} {g(x,y) sqrt {1 - ((k y)/(x^((1/3))))^2))

szklanovqq 2022-11-19 Answered
Completing partial derivatives to make them converge
For a function f ( x , y ) of two independent variables we have an incomplete specification of its partial derivatives as follows:
f ( x , y ) x = 1 g ( x , y ) 1 ( k y x ( 1 / 3 ) ) 2
f ( x , y ) y = ( 3 x 4 ) ( k x ( 1 / 3 ) ) 2 ( 2 y ) 1 g ( x , y ) 1 ( k y x ( 1 / 3 ) ) 2
Problem: finding a suitable g ( x , y ) that makes the partial derivatives converge to a single function f ( x , y ) that fulfills the condition f ( x , 0 ) = x
I will be grateful if people with many flight hours can offer suggestions for g ( x , y ). Needless to say, I am not asking that they verify those suggestions, but in case someone would like, these are the inputs to Wolfram integrator:
1 ( g ( x , y  as  r ) 1 ( k r / x ( 1 / 3 ) ) 2 ) ( 3 t / 4 ) ( k / t 1 / 3 ) 2 ( 2 x ) ( g ( x  as  t , y  as  x ) 1 ( k x / t ( 1 / 3 ) ) 2 )
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Answers (1)

h2a2l1i2morz
Answered 2022-11-20 Author has 19 answers
Hint: From your data one obtains
f x f y = 2 3 k 2 y x 1 / 3 .
It follows that the curves f ( x , y ) =const. satisfy the separable differential equation
y = 2 3 k 2 y x 1 / 3 .
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