Question

Find the general solution of the differential equation: cosx (1 + cosy) dx - siny (1 + sinx) dy = 0

Answer 1

Step 1: Rearrange the equation:
We start with the given equation: \[ \cos(x)(1 + \cos(y)) \, dx = \sin(y)(1 + \sin(x)) \, dy \]

Step 2: Divide both sides by \( \cos(x)(1 + \cos(y)) \):
Rearranging the equation to isolate \( \frac{dx}{dy} \): \[ \frac{dx}{dy} = \frac{\sin(y)(1 + \sin(x))}{\cos(x)(1 + \cos(y))} \]

Step 3: Simplify using trigonometric identities:
To simplify further, we can use the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \). We introduce the identity into the equation to make the terms easier to handle: \[ \frac{dx}{dy} = \frac{\sin(y)(1 + \sin(x))}{\cos(x)(1 + \cos(y))} \times \frac{\sin^2(x) + \cos^2(x)}{\sin^2(x) + \cos^2(x)} \] This results in: \[ \frac{dx}{dy} = \frac{\sin(y)(1 + \sin(x)) \sin^2(x)}{\cos(x)(1 + \cos(y))(\sin^2(x) + \cos^2(x))} \]

Step 4: Further simplification:
After simplifying, we have: \[ \frac{dx}{dy} = \frac{\sin(y) \sin(x) (1 + \sin(x))}{\cos(x)(1 + \cos(y))} \]

Step 5: Separate the variables:
Now, we separate the variables to isolate \( dx \) and \( dy \). Multiply both sides by \( dy \): \[ dy = \frac{\cos(x)(1 + \cos(y))}{\sin(y) \sin(x)(1 + \sin(x))} \, dx \] This simplifies to: \[ dy = \frac{\cos(x)}{\sin(x)} \cdot \frac{1 + \cos(y)}{\sin(y)(1 + \sin(x))} \, dx \]

Step 6: Set up the integral:
We can now integrate both sides of the equation. The left-hand side integrates to \( y \), and the right-hand side involves a more complicated integral: \[ \int dy = \int \frac{\cos(x)}{\sin(x)} \cdot \frac{1 + \cos(y)}{\sin(y)(1 + \sin(x))} \, dx \] Thus, the general solution is: \[ y = \int \frac{\cos(x)}{\sin(x)} \cdot \frac{1 + \cos(y)}{\sin(y)(1 + \sin(x))} \, dx + C \] where \( C \) is the constant of integration.

Final Answer:
Unfortunately, the integral on the right-hand side does not have a simple closed-form solution. Therefore, the general solution of the given differential equation is: \[ y = \int \frac{\cos(x)}{\sin(x)} \cdot \frac{1 + \cos(y)}{\sin(y)(1 + \sin(x))} \, dx + C \] where \( C \) is the constant of integration.