Px(z - 2y ^ 2) = (z - qy)(z - y ^ 2 - 2x ^ 3) find the general integral of pde by Lagrange method?

Question

Answer

General integral of PDE by Lagrange method

Given PDE: px(z - 2y^2) = (z - qy)(z - y^2 - 2x^3)

Explanation:

Step 1: Write the Lagrange multiplier equation

Lagrange method involves introducing a Lagrange multiplier λ to the given PDE. The Lagrange multiplier equation is formed by equating the partial derivatives of the given equation with respect to x, y, and z to the partial derivatives of the unknown function φ with respect to x, y, and z multiplied by λ.

Step 2: Solve the Lagrange multiplier equation

By solving the Lagrange multiplier equation, we can determine the unknown function φ. This involves integrating the partial derivatives of φ with respect to x, y, and z with respect to x, y, and z respectively.

Step 3: Verify the solution

Once the unknown function φ is determined, substitute it back into the original PDE to verify if the solution satisfies the given equation.

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