Question

The complete integral of partial differential equations√p+√q=1 is

Answer 1

Step-by-step explanation:

Given The complete integral of partial differential equations√p+√q=1 is

• Let f(p.q) = √p + √q – 1 = 0 ------------------1
• Now let the complete integral of equation 1 be  
•       So p = ax + by + c ------------------2
• Now f (a,b) = √a + √b – 1 = 0
•       So √b = 1 - √a
•       Or b = (1 - √a)^2
• So the required complete integral is given by
•         So p = ax + (1 - √a)^2y + c

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Answer 2

SOLUTION

TO DETERMINE

The complete integral of

$\sf{ \sqrt{p} + \sqrt{q} = 1 \: }$

EVALUATION

Here the given equation is

$\sf{ \sqrt{p} + \sqrt{q} = 1\: } \: \: .....(1)$

This equation is of the form f(p, q) = 0

So the solution is given by

$\sf{z = ax + by + c} \: \: \: ......(2)$

Where a, b, c are constants

Now in order to get the complete integral we have to eliminate any one of the arbitrary constants.

Differentiating partially both sides of equation (2) with respect to x we get

$\displaystyle \sf{ \frac{ \partial z}{ \partial x} = p = a \: }$

Again Differentiating partially both sides of Equation (2) with respect to y we get

$\displaystyle \sf{ \frac{ \partial z}{ \partial y} = q = b \: }$

Putting these values in Equation (1) we get

$\sf{ \sqrt{a} + \sqrt{b} = 1\: } \: \:$

$\implies \sf{b = {(1 - \sqrt{a} )}^{2} \: }$

Hence the required complete integral is

$\sf{z = ax + {(1 - \sqrt{a})}^{2}y + c }$

Where a and c are arbitrary constants

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