Question

2. Find the complete integral of the partial differential

equation p+q=sin x+sin y.

Answer 1

The given differential equation can be written as

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

Let

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

We have

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

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

Integrate

$$z=ax-\cos x-\cos y-ay+b$$

The complete integral of the given PDE is

$$z=a(x-y)-(\cos x+\cos y)+b$$