Question #316062

Solve the PDE whose auxiliary equations as follows: 𝑑𝑥/ 2𝑦(𝑧 − 3) = 𝑑𝑦 /𝑦(2𝑥 − 𝑧) = 𝑑𝑧 /𝑦(2𝑥 − 3)


1
Expert's answer
2022-03-29T11:44:37-0400

The auxiliary equations is:

dx2y(z3)=dyy(2xz)=dzy(2x3)\frac{dx}{2y(z-3)}=\frac{dy}{y(2x-z)}=\frac{dz}{y(2x-3)}

A first characteristic equation comes from

dx2y(z3)=dzy(2x3)\frac{dx}{2y(z-3)}=\frac{dz}{y(2x-3)}

(2x3)dx=2(z3)dz(2x-3)dx=2(z-3)dz

x23x+94=z26z+9+C1x^2-3x+\frac94=z^2-6z+9+C_1

(x32)2=(z3)2+C1(x-\frac32)^2=(z-3)^2+C_1

C1=(x32)2(z3)2C_1=(x-\frac32)^2-(z-3)^2

A second characteristic equation comes from

dzdyy(2x32x+z)=dx2y(z3)\frac{dz-dy}{y(2x-3-2x+z)}=\frac{dx}{2y(z-3)}

2d(zy)=dx2d(z-y)=dx

2(zy)=x+C22(z-y)=x+C_2

C2=2z2yxC_2=2z-2y-x.

General solution of the PDE on the form of implicit equation:

Φ(C1,C2)=0\Phi(C_1,C_2)=0


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS