Question

Find complete integral of the differential equation :
zpq = p + q.
​

Answer 1

Answer:

The complete integral is ∫ dz Q ( a, z ) =ax+y+b. Example 3.15 Find a complete integral of zpq-p-q=0 → Putting p=aq in the equation, we get zaq-a-1=0 [ or p=0,q=0 in which case z is constant]. If zaq-a-1=0, then q= 1 + a az and p= 1 + a z

Answer 2

Concept:

Complete integrals are partial differential equations of the first order that include as many arbitrary constants as independent variables. They are the solution to these partial differential equations. A differential equation in arithmetic is an equation that connects the variants of one or more unknown functions. Applications often involve functions that reflect physical quantities, derivatives that depict the rates at which those values change, and a differential formula that establishes a connection between the three.

Given:

Here we have been given that the differential equation is $zpq = p + q.$

Find:

We have to find the complete integral of the differential equation.

Solution:

According to the question,

In this case we consider the pde of the form

$f(p,q)=0$

Now, Chaript's auxiliary equations are $dx/f_p=dy/f_q$

$-dp/0=-dq/0$

Now, $dp=0$ and $dq=0$

So, $p$ and $q$ are constant.

Now, we get $dz=pdx+qdy$

If we consider $p=x$and $q=y$ then the integral is $z=ax+by+c$.

Hence, we have done our complete integral of the differential equation and this is our answer.

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