I was solving a physics problem and the solution involves solving a differential equation that's of the form: $$ \bigg ( \frac {dx} {dt} \bigg)^2 + \bigg ( \frac {dy} {dt} \bigg)^2 = a^2 $$ By guessing, ...
Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complic...
Is there a reason it is so rare we can solve differential equations?
The right question is not "What is a differential?" but "How do differentials behave?". Let me explain this by way of an analogy. Suppose I teach you all the rules for adding and multiplying rational numbers. Then you ask me "But what are the rational numbers?" The answer is: They are anything that obeys those rules. Now in order for that to make sense, we have to know that there's at least ...
Then one thinks of differential operators as a linear maps between such spaces. Often the space of all linear maps between two spaces is itself a vector space and so one can indeed start to manipulate differential operators as if they are ‘objects’ in their own right eg add them together.
Why can we treat differential operators as if they behave like ...
Anyone who sees calculus in application is likely to encounter both derivatives and differentials. The two concepts have confusingly similar notation. For that reason, this post is a very important contribution.
calculus - What is the practical difference between a differential and ...
I am a bit confused about differentials, and this is probably partly due to what I find to be a rather confusing teaching approach. (I know there are a bunch of similar questions around, but none o...