The function z is related to x and y such that 6y Find zxy

The function z is related to x and y such that = 6y. Find z(x,y) if at x = 1, = 10 y+5 and z= 13y+12. The function z is related to x and y such that = 6x + 4. Find z(x,y) if at y=2,= 12x+13 and at x= 1 z = 13y+5.

Solution

1.
d²z/dx² = 6y
partially integrate wrt x
dz/dx = 6yx + C..............(1)
given that at x = 1, dz/dx = 10y + 5
10y + 5 = 6y + C
C = 4y + 5
So eq(1) becomes
dz/dx = 6yx + 4y + 5
Again partially integrate wrt x
z = 3yx^2 + 4yx + 5x + C.........(2)
given that at x = 1, z = 13y + 12
13y + 12 = 3y + 4y + 5 + C
C = 6y + 7
Hence equation (2) becomes
z(x, y) = 3yx^2 + 4yx + 5x + 6y + 7

2.
d²z/dydx = 6x + 4
partially integrate wrt y
dz/dx = 6xy + 4y + C............(1)
given that at y = 2, dz/dx = 12x + 13
12x + 13 = 12x + 8 + C
C = 5
dz/dx = 6xy + 4y + 5
Again partially integrate wrt x
z(x, y) = 3x^2y + 4xy + 5x + C
given that at x = 1, z = 13y + 5
13y + 5 = 3y + 4y + 5 + C
C = 6y
z(x, y) = 3x^2y + 4xy + 5x + 6y