An integration or a differential equation not easy to carry out are because many reasons, first the equation includes many independent arguments so the equation not easy to handle and can not figure out a simplified one, sometimes the equation include the dependent function in both sides dazzling in all simplified step usually in many formats and not to mention iteration and error computation are not easy at all when it diverse. When it diverse there is no way to catch a solution, the new way solve this
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Design, the very important verbal phase in manufacturing, to do so you walk from the physics phase then the mathematics phase then go between the both end up with the model, of course you meet such an integration or a differential equation or more of which can easy carry out or not.
Let us talk about those not easy to carry out for example the equation includes the dependent function itself in such case a lot of things need to be done, numerical models and error handling during computing after building the programs related.
These things is a case picture dependent, i.e. related to certain arguments values i.e. the total work ends up with a single value for the argument under conditions, designer can not make verbal induction changes on the argument to any direction.
The designer carry this many times using programming and have a table with the value of the argument, a picture which not clear, what if there are many arguments, the picture becomes not clearer, the designer without a formula obligated to look to all these value tables to imagine a change also when he wants to talk about.
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the new way is simply makes things happen, in the past all ended and stopped at numerical whatever it is needed to continue induction or not, now there is formala able to continue induction or to proceed and have a new formula, i.e. there is no limits induction, derivation, make target’s profile, make sighting and understanding, and talk about it in general, solving a differential equation or integration is essential when encounter and meet it, the story of numerical, iteration and error analysis has to replace with appropriate in order to have a configured formula
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One known case is the differential equation second order second degree let say (d2y/(dx1dx2))2+4×1=0 or differential equation second order second degree self dependence (d2y/(dx1dx2))2+sin(y2)+2(y2)3=0, both are solvable numerically, the first is solvable analytically i.e. could have an exact solution but with the second equation only a value as a result of the numerical is the solution. The first designer with first equation can draw profiles and make induction and make appropriate move toward his target where the second designer with the second equation cannot, what if the first designer was with the second equation, he cannot make the appropriate moves too. Second example of difficulty
(d2y/(dx1dx2))2/(dy/dx3)+4×1=0, in the first term of the equation the enumerator and the denumerator are differential of the dependent function
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As an important example, is the differential equation fifth order third degree, we want here to distinguish between two fundamentals first the differential equation is easy let say we have three independents namely x1,x2,x3, (d5y/dx1dx2(d3x3))3+x1=0, the second with the same independents (d5y/dx1dx2(d3x3))3/(d3y/d3x3)+len(y)-4y3=0 which is not easy, iteration is needed with error handling computed many times numerically to figure out values not even enough to know any profile and we have a nonfigurative attribution case
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A simple example of related, the two cases, first the integration y=Ssin(x)cos(x)dx which is easy having the exact solution y=f(x) of course able to make insights about f(x) and the second y=Ssin(x)cos(y)dx which is not easy; actually you need to walk from and back between iteration and management error and compute many times numerically making the required computer programs too and at the end you have the appropriate table of data using it carefully, these tables, to figure out an insight to make a plan out of a group of plans to make the figured out change of the argument
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An integration or a differential equation not easy to carry out are because many reasons, first the equation includes many independent arguments so the equation not easy to handle and can not figure out a simplified one, sometimes the equation include the dependent function in both sides dazzling in all simplified step usually in many formats, i.e. the equation strongly dependent on itself, and not to mention iteration and error computation are not easy at all when it diverse. When it diverse there is no way to catch a solution
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Attention this is not a repeated or usual talk, in engineering design cases walk from phase to phase forward and backward and during this a lot of work is required before settlement with a differential equation or an integration equation which describe such case, being such equation is not solvable but numerically of course that with a lot of precautions is not the destination which was waited for by all costs.
Numerical solution seems to be not the appropriate one even if it is doable and the only one left so far because the loss of the potential of features and attributes with which the missing exact solution formula describes the case physical dependence and the afterward mathematical model, a typical vague picture, things get more vaguely if the equation is a function of itself, self dependent, i.e. function of the dependent variable the equation would have meant to be solved for also there some challenges like where there is a differentiation in a denumerator in the differential equation; this vaguely situation decreases the power of the tools used in design dramatically, all this because there is no a formula describe such.