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Initial Condition(s) are a set of conditions, or a condition on the solution which will permit us to find out that solution which we are after. Initial conditions are frequently abbreviated of the form:
y(t_{0}) = y_{0} and or y^{(k)}(t_{0}) = y_{k}
Thus, we can say, initial conditions are values of the solution or/and its derivative(s) at exact points. As we will notice eventually, solutions to "nice sufficient" differential equations are unique and thus only one solution will meet the specified conditions.
For a specified differential equation the number of initial conditions are needed, will depend on the order of the differential equation as we will notice.
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