Question

Solve the differential equation \(\sin x \cos y\,dx + \cos x \sin y\,dy = 0\), given that \(y = \frac{\pi}{4}\) when \(x = 0\).

Hint:

Divide by cos x cos y to get tan x dx + tan y dy = 0, integrate, then use y = pi/4 at x = 0.

Answer 1

Step 1: Separate the variables.
Divide every term by \(\cos x \cos y\). This is allowed where both cosines are non-zero.
\[ \frac{\sin x}{\cos x}\,dx + \frac{\sin y}{\cos y}\,dy = 0 \implies \tan x\,dx + \tan y\,dy = 0 \]

Step 2: Integrate both sides.
We use \(\int \tan x\,dx = -\log|\cos x|\).
\[ -\log|\cos x| - \log|\cos y| = -\log|C| \]
Multiply by \(-1\) and combine the logarithms.
\[ \log|\cos x \cos y| = \log|C| \implies \cos x \cos y = C \]
This is the general solution.

Step 3: Use the given condition.
Put \(x = 0\) and \(y = \frac{\pi}{4}\).
\[ C = \cos 0 \cdot \cos\frac{\pi}{4} = 1 \cdot \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

Step 4: Write the particular solution.
Put this value of C back in the general solution.

Final Answer:
The particular solution is the following. \[ \boxed{\cos x \cos y = \frac{1}{\sqrt{2}}} \]