The complete solution of the partial differential equation q (p – cos x) = cos y is

Question

Question

Answer 1

Concept:

Separable equation of the form f(x, p) = g(y, q)

Let f(x, p) = g(y, q) = a (constant)

Solve f(x, p) = a and g(y, q) = a for p and q, we get

p = ϕ(x, a) and q = ψ(y, a)

We have dz = p dx + q dy

Bu integration, we get the required solution as follows,

z = ∫ ϕ(x, a) dx + ∫ ψ(y, a) dy + b

Where a, b are arbitrary constants.

Calculation:

Given PDE is q (p – cos x) = cos y;

Rearranging, we get

p – cos x = (cos y)/q;

Lets assume p – cos x = (cos y)/q = a

⇒ p = a + cos x

\( \Rightarrow q = \frac{{\cos y}}{a}\)

We have dz = p dx + q dy

\( \Rightarrow dz = \left( {a + \cos x} \right)dx + \frac{{\cos y}}{a}dy\)

\(\Rightarrow z = ax + \sin x + \frac{{\sin y}}{a} + b\)