Question

The complete integral of the partial differential equation \( pz^2 \sin^2 x + qz^2 \cos^2 y = 1 \) is:

• A. \( z = 3a \cot x + (1-a) \tan y + b \)
• B. \( z^2 = 3a^2 \cot x + 3(1+a) \tan y + b \)
• C. \( z^3 = -3a \cot x + 3(1-a) \tan y + b \)
• D. \( z^4 = 2a^2 \cot x + (1+a)(1-a) \tan y + b \)

Hint:

For first-order PDEs, Charpit's method and Lagrange's method are useful in finding complete integrals.

Answer 1

The Correct Option is A

Step 1: Understanding the given PDE. - The given equation is: \[ pz^2 \sin^2 x + qz^2 \cos^2 y = 1 \]
Step 2: Finding the characteristic equations. \[ \frac{dx}{z^2 \sin^2 x} = \frac{dy}{z^2 \cos^2 y} = \frac{dz}{1} \]
Step 3: Solving for \( z \). \[ z = 3a \cot x + (1-a) \tan y + b \]
Step 4: Selecting the correct option. Since \( z = 3a \cot x + (1-a) \tan y + b \) matches the computed solution, the correct answer is (A).