Exact Differential Equations – Differential Equations of First Order and First Degree


Hey students so today we are gonna learn very interesting topic of mathematics which is called as exact differential equation now let’s first understand what is exact differential equation and how to find out the solution of exact differential equation because whenever we are gonna solve the differential equation we might come across the case where we will get the exact differential equation so for that you must know that how to identify that exact differential equation and next after identifying how to solve that equation to get the solution so guys here we have the definition of exact differential equation so any differential equation of the form M DX plus n dy equal to 0 where m and n function of x and y or can be constant so here M DX plus n dy equal to 0 is a differential equation it says some differential equation and it is said to be exact provided it is satisfying one condition that is dou M upon dou Y is equal to dou n upon dou X now guys what is dou M upon the way so this symbol denotes the partial differentiation so whenever the partial differentiation of M with respect to Y is equal to partial differentiation of n with respect to X then that time we say that the given differential equation is exact so this is a way to identify the exact differential equation so whenever we are going to get any equation we are going to check a condition of doing upon dou Y whether it is equal to dou n upon dou X or not if it is safe then we will say that yes it is exact differential equation now guys once we written define that the given equation is the exact differential equation then the next task is to solve that equation because in the examination we always get a question that solve differential equation so what is the solution of exact differential equation so the solution of exact differential equation can be given in one of the two formats so I’ll show you good the solutions so this is the first solution which says it is integration of M DX now whenever we are integrating M I as I said that M is a function of x and y so in that what we have to do is we have to treat Y is constant so we will be having terms of X terms of Y we will treat Y is constant and whatever are the terms of X we will integrate it with respect to X plus integration of n and inside that and we have to consider the terms which are free from X so again n is function of x and y so we just have to consider the terms which are free from X so it means we want terms which has only Y okay so we are going to integrate those terms with respect to Y and addition of that equal to C is nothing but our solution so guys this is a way to find out the solution of exact differential equation now many times it happens that we are not able to find out the solution by using this first formula so in such cases what do we do we use the second formula so what is the second formula and second formula we are just exchanging the terms so for M I am replacing it with N and that n is being replaced by M so here we say we have to integrate n with respect to Y okay you have to keep this in mind because when we were integrating M with respect to X that time N was being integrated by or with respect to Y similarly here we will integrate n with respect to Y only but here that time we will treat the x-terms constant because n has X as well as Y terms similarly in the second term we will integrate M but from M we will just take the terms which are free from Y so we will integrate those terms with respect to X equal to C can be the solution of exact differential equation so guys here we have seen the two different types of solution for exact differential equation and now in the further videos we are gonna solve the differential equations or I would say the exact differential equation by using this formula okay so I hope you liked this video and you want to see more videos on exact differential equations so don’t worry you have to just log in to each a.com today itself where you will find all videos of engineering mathematics and your favorite topic that is differential equation and you can access all the videos at one place thank you very much signing off professor fish

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