Answer to Question #98886 in Differential Equations for SGanguly

Question #98886
Find the complete integral using Charpit Method : 2x(q²z²+1) = pz
1
Expert's answer
2020-07-30T15:27:10-0400


Given : "f(x,y,z,p,q)=2xq^2z^2+2x-pz"

Now we use Charpit's method, for which the following integrals are needed:

"\\partial f\/\\partial x=2q^2z^2+2"

"\\partial f\/\\partial y=0"

"\\partial f \/\\partial \/z=4xq^2z-p"

"\\partial f\/\\partial p=-z"

"\\partial f\/\\partial q=4xqz^2"


Clearly, we take the easiest two terms from Charpit's equation; we get;

"dx\/z =dy\/(-4xqz^2)"

Solving this we get;

"-\\int 4xqzdx=\\int dy"

"\\implies -2qzx^2=y+a"

"\\implies q=-(y+a)\/(2x^2z)" , where "a" is a constant of integration


Putting this value of "q" in "f(x,y,z,p,q)" we get;

"pz=2xz^2[-(y+a)\/2x^2z]^2+2x"

"\\implies p=2x\/z + [(y+a)^2\/2x^3z]"


Now we put these obtained values in : "pdx+qdy=dz"

"dz=(2x\/z+[(y+a)^2\/2x^3z])dx - ((y+a)\/2x^2z"")dy"

"zdz=2xdx+1\/4d((y+a)^2\/x^2)"


Integrating we get:

"z^2\/2=x^2+[(y+a)^2\/4x^2]+b" (Answer)

where "a" and "b" are constants of integration.



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Comments

Assignment Expert
03.02.21, 19:53

Dear AKASH.R, You are welcome. We are glad to be helpful. If you liked our service, please press a like-button beside the answer field. Thank you!

AKASH.R
03.02.21, 06:46

It's very good

Assignment Expert
30.07.20, 00:46

It is possible to solve other partial differential equations, for example, 2x(q^2z^2+1)=a, pz=a, where a is a constant.

rahul
29.07.20, 22:10

how can we integrate (4xqz dx) taking z as constant if it is dependent on x ?

Assignment Expert
20.11.19, 19:28

Please kindly wait for a solution of this question.

SGanguly
20.11.19, 04:56

It has been days still no response

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