Answer to Question #5983 in Differential Equations for Akhtar Rasoolddd

Question

Dear Expert Dr math i have one problem please try to explain.Thank you
what is initial value problems and boundary vale problems.

Answer 1

In mathematics, in the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.

A simple example is to solve y' = 0.85y and y(0) = 19. We are trying to find a formula for y(t) that satisfies these two equations.

A boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.

Concretely, an example of a boundary value (in one spatial dimension) is the problem
y"(x) + y(x) = 0
to be solved for the unknown function y(x) with the boundary conditions
y(0) = 0, y(pi/2) = 2.