- #1
matpo39
- 43
- 0
hi, I am having some trouble with this problem:
A certain function f(x,y) is known to have partial derivatives of the form
(partial)f/(partial)x = 2ycos(2x)+y^3*x^2+g(y)
(partial)f/(partial)y= sin(2x)+x^3*y^2+4x+1
Please note that g is a function of y only. Use the equality of mixed partial derivatives (Clairaut's Theorem) to find the function g up to an arbitrary additive constant. then find all the functions f.
i was able to attempt the first part and i got
g'(y)= 4
and then intagrating that i find that g(y)= 4y+c_1
i then inteagrated the two partial functions and i got
f(x)= ysin(2x)+(1/3)y^3*x^3+4yx+c_2
f(y)= ysin(2x)+(1/3)y^3*x^3+4yx+y+c_3
i have no idea on what i should do next, if anyone can help me out, that would be great.
thanks
A certain function f(x,y) is known to have partial derivatives of the form
(partial)f/(partial)x = 2ycos(2x)+y^3*x^2+g(y)
(partial)f/(partial)y= sin(2x)+x^3*y^2+4x+1
Please note that g is a function of y only. Use the equality of mixed partial derivatives (Clairaut's Theorem) to find the function g up to an arbitrary additive constant. then find all the functions f.
i was able to attempt the first part and i got
g'(y)= 4
and then intagrating that i find that g(y)= 4y+c_1
i then inteagrated the two partial functions and i got
f(x)= ysin(2x)+(1/3)y^3*x^3+4yx+c_2
f(y)= ysin(2x)+(1/3)y^3*x^3+4yx+y+c_3
i have no idea on what i should do next, if anyone can help me out, that would be great.
thanks