How do you use separation of variables method to find the solution of the following differential equation?

Question

#4(du)/dx -3(du)/dy = 0 #

Given #u(0,y) = 16e^(4y)#

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Answer 1 · 2018-04-18T08:14:11

#u(x,y) = 16 e^(3x + 4y) #

#u(x,y) = xi(x) zeta(y)#

#u_x = xi' zeta# and #u_y = xi zeta'#

#4 u_x -3 u_y = 0 implies 4 (xi')/xi =3 (zeta')/ zeta = C#

# implies xi(x) = alpha e^((C x)/4) # and # zeta(y) = beta e^((C y)/3)#

So: #u(x,y) = gamma e^(C (x/4 + y/3)) #

Applying the IV:

#u(o,y) = gamma e^(C y/3) = 16 e^(4y) implies (gamma, C) = (16, 12)#

So: #u(x,y) = 16 e^(3x + 4y) #