Answer to Question #262903 in Calculus for Unknown346307

Question #262903

Find a complete and singular integrals of 2xz − px2 − 2qxy + pq = 0


1
Expert's answer
2021-11-09T14:19:10-0500

Solution;

"f(x,y,z,p,q)=2xz-px^2-2qxy+pq=0.....(1)"

Now the charpits auxiliary equations are;

"\\frac{dp}{2z-2qy}=\\frac{dq}{0}=\\frac{dz}{px^2-pq+2xyq-pq}=\\frac{dx}{x^2-q}=\\frac{dy}{2x-p}"

The second fraction gives dq=0 ,which implies that q=a,a being an arbitrary constant.substitute in (1) we have;

"2zx-px^2-2axy+pa=0"

Which gives;

"p=\\frac{2x(z-ay)}{x^2-a}"

Substitute the values of p and q in "dz=pdx+qdy" we get;

"dz=\\frac{2x(z-ay)}{x^2-a}dx+ady"

Rewritten as;

"\\frac{dz-ady}{z-ay}=\\frac{2x}{x^2-a}dx"

After integration;

"log(z-ay)=log(x^2-a)+logb"

Or

"(z-ay)=b(x^2-a)"

From which;

"z=ay+b(x^2-a)"

(a and b are arbitrary constants)


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!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS