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Answer to Question #5983 in Differential Equations for Akhtar Rasoolddd

Question #5983
Dear Expert Dr math i have one problem please try to explain.Thank you
what is initial value problems and boundary vale problems.
Expert's answer
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.

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