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)=2xq2z2+2xpzf(x,y,z,p,q)=2xq^2z^2+2x-pz

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

f/x=2q2z2+2\partial f/\partial x=2q^2z^2+2

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

f//z=4xq2zp\partial f /\partial /z=4xq^2z-p

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

f/q=4xqz2\partial f/\partial q=4xqz^2


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

dx/z=dy/(4xqz2)dx/z =dy/(-4xqz^2)

Solving this we get;

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

    2qzx2=y+a\implies -2qzx^2=y+a

    q=(y+a)/(2x2z)\implies q=-(y+a)/(2x^2z) , where aa is a constant of integration


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

pz=2xz2[(y+a)/2x2z]2+2xpz=2xz^2[-(y+a)/2x^2z]^2+2x

    p=2x/z+[(y+a)2/2x3z]\implies p=2x/z + [(y+a)^2/2x^3z]


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

dz=(2x/z+[(y+a)2/2x3z])dx((y+a)/2x2zdz=(2x/z+[(y+a)^2/2x^3z])dx - ((y+a)/2x^2z)dy)dy

zdz=2xdx+1/4d((y+a)2/x2)zdz=2xdx+1/4d((y+a)^2/x^2)


Integrating we get:

z2/2=x2+[(y+a)2/4x2]+bz^2/2=x^2+[(y+a)^2/4x^2]+b (Answer)

where aa and bb 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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