Question

Solve the PDE (mz-ny)p+(nx-lz)q = (ly-mx)

Answer 1

Answer:

Step-by-step explanation:Given equation is Lagranges equation

Pp+Qq=R

The auxiliary equation is

dx

mz−ny

=

dy

nx−lz

=

dz

ly−mx

Each ratio is equal to

ldx+mdy+ndz

l(mz−ny)+m(nx−lz)+n(ly−mx)

=

ldx+mdy+ndz

0

⇒ldx+mdy+ndz=0

Integrating, lx+my+ny=a

Also, each ratio is equal to

xdx+ydy+zdz

x(mz−ny)+y(nx−lz)+z(lx−my)

=

xdx+ydy+zdz

0

⇒xdx+ydy+zdz=0

Integrating,

x2

2

+

y2

2

+

z2

2

=c1

⇒x2+y2+z2=b

∴ The general solution is

ϕ(lx+my+nz,x2+y2+z2)=0