Ordinary Differential Equations/Exact equations

IntroductionEdit

Suppose the function   represents some physical quantity, such as temperature, in a region of the  -plane. Then the level curves of F, where  , could be interpreted as isotherms on a weather map (i.e curves on a weather map representing constant temperatures). Along one of these curves,  , of constant temperature we have, by Chain rule and the fact that the temperature, F, is constant on these curves:

 

Multiplying through by   we obtain

 
Therefore, if we were not given the original function F but only an equation of the form:

 

we could set   and then by integrating figure out the original  .

Method formal stepsEdit

(1) First ensure that there is such an  , by checking the exactness-condition:

 

This is because if there was such an F, then  

where   and   simply denote the partial derivatives with respect to the variables   and   respectively (where we hold the other variable constant while taking the derivative).

(2)Second, integrate   with respect to   respectively:

 
 

for some unknown functions   (these play the role of constant of integration when you integrate with respect to a single variable). So to obtain   it remains to determine either   or  .

(3)Equate the above two formulas for  : 

(4) Since to find   it suffices to determine   or  , pick the integral that is easier to evaluate. Suppose that   is easier to evaluate. To obtain   we differentiate both expression for   in   (for fixed  ): and then integrate in  : 

(5)Observe that   is only a function of   since if we differentiate the expression we found for   and use step   we find that