Answer to Question #223889 in Chemical Engineering for Lokika

Question

b) Find the complete integral of the PDE px + qy +z = xq^2?

Answer 1

dp/(2p+2p) = dy/-2y

dp/p + 2*dy/y =0

py^2=a

Put the value of y = a/y^2 in (1)

q = - z/y - ax/y^3 + a^2 / (2y^4)

Now, complete solution can be found by the following equation

dz = pdx + qdy

dz = (a/y^2) dx - (z/y) dy - (ax/y^3)dy + (a^2/(2y^4)) dy

ydz + zdy = a((ydx - xdy) /y^2) + (a^2 / (2y^3)) dy

yz - ax/y +(a^2) /(4y^2) = b

z = ax/y^2 - a^2 / (4y^3) + b/y