Partial Differential Equations: Constant Elimination

• Partial Differential Equations (PDEs) involve functions of several variables and their partial derivatives.

Eliminating Arbitrary Constants

• The goal is to form a PDE that does not contain the arbitrary constants present in a given relation.
• Start with a relation involving independent variables (e.g., x, y), a dependent variable (e.g., z), and arbitrary constants (e.g., a, b).
• Differentiate the given relation with respect to each independent variable.
• The number of differentiations needed usually corresponds to the number of arbitrary constants.
• Eliminate the arbitrary constants from the original relation and the derivatives obtained.
• The resulting equation will be the required PDE.

Example: Eliminating a Single Arbitrary Constant

• Given: z = ax + by
• Differentiate with respect to x: ∂z/∂x = a
• Differentiate with respect to y: ∂z/∂y = b
• Substitute a and b back into the original equation to get the PDE: z = x(∂z/∂x) + y(∂z/∂y)

Example: Eliminating Two Arbitrary Constants

• Given: z = ax + by + ab
• Differentiate with respect to x: p = ∂z/∂x = a
• Differentiate with respect to y: q = ∂z/∂y = b
• Substitute a and b into the original equation: z = px + qy + pq, which is the required PDE.

Dealing with More Constants

• With more constants, more differentiation steps and algebraic manipulation are usually needed.
• The process may involve solving systems of equations to eliminate the constants.
• The order of the resulting PDE depends on how many times you differentiate, but it relates to the number of independent variables and arbitrary constants.

Eliminating Arbitrary Functions

• Objective: To form a PDE that does not contain the arbitrary functions present in a given relation.
• Start with a relation involving independent variables (e.g., x, y), a dependent variable (e.g., z), and arbitrary functions (e.g., f, g).
• Differentiate the relation with respect to each independent variable.
• The number of differentiations depends on the number and type of arbitrary functions.
• Eliminate the arbitrary functions and/or their derivatives from the original relation and the derivatives obtained.
• The result will be the required PDE.

Example: Eliminating a Single Arbitrary Function

• Given: z = f(x + y)
• Let u = x + y, so z = f(u)
• Differentiate z with respect to x: ∂z/∂x = f'(u) * (∂u/∂x) = f'(u) * 1 = f'(u)
• Differentiate z with respect to y: ∂z/∂y = f'(u) * (∂u/∂y) = f'(u) * 1 = f'(u)
• Therefore, ∂z/∂x = ∂z/∂y or p = q, which is the required PDE.

Example: Eliminating a Different Form of Arbitrary Function

• Given: z = xf(x/y)
• Differentiate z with respect to x: p = ∂z/∂x = f(x/y) + x * f'(x/y) * (1/y)
• Differentiate z with respect to y: q = ∂z/∂y = x * f'(x/y) * (-x/y^2)
• Multiply q by y: yq = x * f'(x/y) * (-x/y)
• Compare with p: p = f(x/y) + x * f'(x/y) * (1/y)
• Note that x * f'(x/y) * (1/y) = -yq/x
• Substitute this into the p equation: p = f(x/y) - yq/x
• Multiply by x: px = xf(x/y) - yq
• Since z = xf(x/y), then px = z - yq
• Rearrange: px + yq = z

Cases with Multiple Arbitrary Functions

• When multiple arbitrary functions are involved, the elimination process can become more complex.
• The derivatives of the functions may also need to be eliminated.
• Some standard techniques involve clever differentiation and algebraic manipulation to isolate and eliminate the arbitrary functions.

General Strategy

• Identify the arbitrary constants or functions in the given relation.
• Determine the number of differentiations needed with respect to the independent variables. This often corresponds to the number of constants or nature of the functions.
• Perform the differentiations.
• Algebraically manipulate the equations to eliminate the arbitrary constants or functions.
• Express the final result as a PDE involving the dependent variable and its partial derivatives.