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If a differential equation does have a solution how many solutions are there?
As we will see ultimately, this is possible for a differential equation to contain more than one solution. We would like to identify how many solutions there will be for a specified differential equation.
A sub-question here is as well. What types of conditions on a differential equation are required to acquire a single unique solution to the differential equation?
Both such question and the sub-question are more significant than you might realize. Assume that we derive a differential equation which will provide the temperature distribution in a bar of iron at any time t. If we resolve the differential equation and end up along with two or more completely different solutions we will have problems. Look at the subsequent situation to see this.
If we subject 10 identical iron bars to identical conditions they must all exhibit similar temperature distribution. Thus only one of our solutions will be correct, but we will have no technique of knowing that one is the accurate solution.
This would be nice if, throughout the derivation of our differential equation, we could ensure that our assumptions would provide us a differential equation by solving, will yield a particular unique solution.
This question is generally termed as the uniqueness question in a differential equations course.
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