Question

The complete integral of the following pq =1​

Answer 1

Answer:

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Answer 2

The complete question is "Find the complete integral of pq=1 by charpit's method"

Answer

The complete integral of pq=1 by Charpit's method is $z=ax+\frac{ay}{3a-1}+b$

Given:

The equation pq=1

To find:

The complete integral of pq=1 by Charpit's method.

Step-by-step explanation:

Solve the expression $f(x, y, z, p, q)=p+q-3pq =0 }$                (1)

Charpit's auxiliary equations for above expression (1) is

$\frac{d p}{\frac{\delta f}{\delta x}+p \frac{\delta f}{\delta z}}=\frac{d q}{\frac{\delta f}{\delta y}+q \frac{\delta f}{\delta z}}=\frac{d z}{-p \frac{\delta f}{\delta p}-q \frac{\delta f}{\delta q}}=\frac{d x}{\frac{-\delta f}{\delta p}}=\frac{d y}{\frac{\delta f}{\delta q}}$

$\left.\frac{d p}{0+p .0}=\frac{d q}{0+q .0}+\frac{d z}{-p(1-3 q)-q(1-3 p)}=\frac{d x}{3 q-1}=\frac{d y}{3 p-1}$                 (2)
By taking the first fraction of (2), we obtain dp = 0

By integrating it, we get p = a

ubstituting the value  p = a in (1), we get

$q=\frac{a}{3 a-1}$

Now, putting the values of p and q respectively from (3) and (4) in $dz=p dx+q dy$,

$dz=a dx+\frac{a}{3a-1}dy$

Integrating it, we obtain $z=ax+\frac{ay}{3a-1}+b$

Hence, the complete integral of pq=1 by Charpit's method is $z=ax+\frac{ay}{3a-1}+b$$z=ax+\frac{ay}{3a-1}+b$