initial conditions lead to linear systemsImposing initial conditions on linear DEs leads to linear systems of equations to solve for the arbitrary cconstants. Here is a second order example, with two initial conditions.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 can visualize this solution by graphing it.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 will learn how to solve such a DE later. For now we want to develop the linear algebra matrix tools to attack such linear systems "systematically"! :-)For linear DEs, imposing initial conditions on the general solution which is itself a linear expression in the arbitrary constants which arise in the solution process leads to a square linear system to solve in order to determine explicit values of those constants. For a single DE there is one linear condition. For a second order DE with two initial conditions and 2 arbitary constants, there is a pair of linear equations to solve, which we can do using high school techniques. However, we want to learn matrix techniques which work for any order linear DE.
For a second order DE we specify the value and derivative of the unknown at some chosen value of the independent variable.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 evaluate (and simplify) these two equations at the chosen initial value of the independent variable, replacing the LHS's by their desired initial values. This leads by a 2x2 system to solve. . We use the percent sign to feed out output into the next input.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