PARTIAL D.E

What is the general form of a first-order PDE?

• u = f(x) + g(y)
• ∂u/∂t + ∂u/∂x = 0
• F(x, y, u, p, q) = 0, where p = ∂u/∂x and q = ∂u/∂y.
• F(x, y, u) = 0

Which of the following is a first-order PDE?

• A first-order PDE is one that involves first derivatives of the unknown function.
• A first-order PDE is defined by its boundary conditions rather than its derivatives.
• A first-order PDE is one that has no derivatives of the unknown function.
• A first-order PDE involves second derivatives of the unknown function.

What is the degree of a PDE?

• The degree of a PDE is the highest power of the highest order derivative in the equation.
• The degree of a PDE is always one.
• The degree of a PDE is the total number of derivatives in the equation.
• The degree of a PDE is determined by the coefficients of the equation.

The order of a PDE is defined as:

• The sum of the coefficients of the derivatives.
• The highest order of the derivative present in the equation.
• The total number of derivatives in the equation.
• The lowest order of the derivative present in the equation.

A PDE is called quasilinear if:

• A PDE is called quasilinear if it has constant coefficients.
• A PDE is called quasilinear if it is nonlinear in all variables.
• A PDE is called quasilinear if it contains no derivatives.
• A PDE is called quasilinear if it is linear in the highest order derivatives.

The term "integral surface" refers to:

• A type of mathematical integral.
• A graphical representation of a single solution.
• A surface that has no solutions.
• A surface representing a family of solutions to a differential equation.

A non-linear PDE can be solved by:

• Various methods including numerical simulations, method of characteristics, and perturbation methods.
• They can only be solved using linear approximations.
• Non-linear PDEs cannot be solved at all.
• Only analytical solutions are valid.

The solution to a linear PDE is:

• A function that satisfies the linear PDE and the associated boundary/initial conditions.
• Any random function without regard to boundary conditions.
• A constant function that does not depend on the variables.
• A function that only satisfies the initial conditions but not the PDE.

A system of PDEs is said to be incompatible if:

• There is a unique solution for each equation.
• The equations can be solved independently.
• There are no solutions that satisfy all the equations simultaneously.
• All equations have at least one solution.

A fundamental solution to a PDE is:

• A solution that only applies to linear equations.
• A method for approximating solutions to PDEs.
• A type of boundary condition used in PDEs.
• A solution that represents the response of a system to a point source.