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Derivatives & Differential Equations |
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The boundary conditions on a differential equation are the constraining values of the function at some particular value of the independent variable. For example, if the equation involves the velocity, the boundary condition might be the initial velocity, the velocity at time t=0. In order to have a complete solution, there must be a boundary condition for each order of the equation - two boundary conditions for a second order equation, only one necessary for a first order differential equation. If a solution to a differential equation is found which satisfies all the boundary conditions, then it is the only solution to that equation - this is called the uniqueness theorem. Therefore, a reasonable approach to finding solutions to differential equations in physical problems is to use a trial solution and try to force it to fit the boundary conditions. If successful, then this approach finds the unique solution. |
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