Question

obtain the complete solution of
p+q = sin x + sin Y

Answer 1

Hi,This is your Answer Happy Learning.

Answer 2

Concept:

A complete integral could be a solution of a partial equation of the primary order that contains as many arbitrary constants as there are independent variables.

Given:

A given equation is $p+q=\sin x+\sin y$.

Find:

We have to seek out the entire solution of the given equation.

Solution:

Firstly, we'll rearrange the given equation by shifting $q$ to the right-hand side and $\sin x$ to the left-hand side of the given equation, we get

$p-\sin x=\sin y-q$

Now, allow us assume $a=p-\sin x=\sin y-q$

Further, we'll simplify the worth of $p$ and $q$ from above, we get

$p=a+\sin x\\q=\sin y-a$

Furthermore, we'll find $dz=pdx+qdy$ by substituting the values of $p$ and $q$ , we get

$dz=(a+\sin x)dx+(\sin y-a)dy$

Now, we'll integrate the above equation, we get

$\begin{aligned}\int dz&=\int (a+\sin x)dx+\int (\sin y-a)dy\\ z&=ax-\cos x-\cos y-ay+C\end$

The above equation can not be integrated more.

Hence, the entire solution of $p+q=\sin x+\sin y$ is $z=ax-\cos x-\cos y-ay+C$.

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