Hello Students Today, I am here with another topic for you all In this video series, we started with first-order differential equation in which my first video was about order and degree and in this, we discussed about two methods variable separable method and homogeneous differential equation of first-order or first degree we also discussed the definition of order and degree there was the second video in which we discussed about linear differential equation and reducible to the linear differential equation what is the definition, how to solve it and based on this we solved questions too Today I am here with my third video the differential equation of the first order and first degree in this, we will discuss what is an exact differential equation how we will solve it and one more topic is there in which we will talk about reducible to the exact differential equation and how to solve such questions so let us see what is an exact differential equation This is the required form of exact differential equation if this is the form of any differential equation if this is equal then the differential equation will be exact differential equation and if in case it is not equal then we will use another method then, students, you will have a question that how we will come to know that it is an exact differential equation or not in that case, in order to solve any question you should see that can we use variable separable method if not then can we use reducible to variable separable method after that, we will see is it a homogeneous equation or not if it is homogeneous you will come to know at first look then we talked about linear differential equation that if in any equation the x is coming once then it id linear in x otherwise it is linear in y in case of exact differential equation here also you will come to know easily as in each question, you will have to calculate this value and see if it is equal or not and if it is equal then the equation is exact now if it is exact how to solve it this is the method to solve it this is the final solution and above two steps is the working process thats how we can solve exact differential equation we have a short cut trick too if we want we can solve this question as we can combine these three steps and here y we have to take as constant when we will do integration, y should be constant here we have, this lengthy method and we can cut short it in this method we will do the direct integration of m but when we will do integration, y must be constant and in this n we will take those terms in which x is not included when we will understand this method while solving questions I will tell you how we will apply it. now we will solve questions on exact differential equation so here we are with 3 questions so let us check, this method is applicable here or not we will solve it as follows we have to do partial differentiation here which means you have to differentiate with respect to y and x should be constant and here we will do it in respect of x and y will be constant as we can see both the values are equal we discussed the working procedure as follows we will differentiate n and put we will do its differentiation with respect to y so this is our required value moving on to third step we will solve it further this is the way we can solve this question let us move on to another question we will proceed as shown here, m and n both are given then we will check it is exact or not so what we will do after differentiating both the values are coming equal which means it is an exact differential equation and this is the method we discussed for solving exact equation now here we are using the short cut method you can use this method in the previous question too. one method we discussed was a bit lengthy one one is a short method, that we will apply in this question I am writing again the short method so students if it is exact then we will directly do its integration you need to be careful while doing integration, here, y will be constant and here, in this n we have to include that term in which there is no x as you can see if we multiply, every term has an x so the value will be 0 in the whole term, the x is involved so it will be 0 now we will do its integration so this is our answer let us see another question of same type here we are given the value of m and n we differentiate it with respect to y and this we will differentiate it with respect to x so the value will be same so this equation is exact same formula we will use it here so let us see how we will do it from this, only those terms that do not include x is to be taken as the whole term contains x, so its value will be 0 we will do its integration and here y will be constant so this is our final answer so student, by using the concept of the exact differential equation we solved three questions here what is exact differential equation and we can solve it in two ways amongst which one is a lengthy procedure and we studied the short cut method and these three questions were based on this concept the next we will discuss, that in case it is not equal then how we will solve it so the method is reducible to exact differential equation As we discussed if any equation is exact we discussed three questions on that part and if by chance a differential equation is not exact so in that case what we will do is, we will check that can we reduce it in an exact differential equation or not? So, here we have 3 methods so, first of all, we have to check that it is not equal so the first case might be that the equation is homogeneous or not if it is homogeneous then we will calculate its integrating factor and by calculating its integrating factor we will reduce it in exact differential equation then we have another form that from m we can take x common and from n, y common then we can calculate its integrating factor and solve it further then the third one is the direct method, which we will discuss later so, first of all, we will discuss about if any differential equation is not exact and it is homogeneous then how we will reduce it into exact this is the required form of reducible to exact differential equation if it is not equal, first we will check it is homogeneous or not if you have mdx and ndy and this equation is homogenous in x and y and it is not equal then this will be its integrating factor. which means we will calculate this value and multiply it with the whole equation so it will be reduced to exact differential equation let us understand this with an example look at the question here and observe how we will solve it as you can see all the terms have degree as 3 so this is a homogeneous differential equation you have to be very careful while taking the values as it is not exact but it is homogeneous so we will calculate its integrating factor first of all, we will calculate the value as shown m will be multiplied by x n will be multiplied by y so our final value will be so the integrating factor will be Now the integrating factor will be multiplied by this equation after multiplying what we will get now we will check it again is it exact or not now we will differentiate it further now both the values are coming equal, so it is an exact differential equation this equation is reduced to exact differential equation Now once it is reduced we will use the short cut formula solve it as follows substitute the values as follows we will further do its integration so this is our final answer this is the way we will solve our question let me explain it again, this is an exact differential equation all the degrees are equal we have checked it and it is not equal we will calculate its integrating factor and this is the formula after multiplying we will check again it is coming equal or not as it is been reduced we will calculate it using the short cut method let us now discuss the second method that it is not exact and also it is not homogeneous then what we will do if it is this form in first y is coming common and then x is coming common so this is the required integrating factor and obviously, it should not be exact let us take two questions here y is coming common and in another x is coming common in this question also, it's the same thing and it is not exact so we will solve it as follows we will differentiate it as you can see the values are not same as it is not exact we will calculate its integrating factor we will multiply m by x solve it as shown its integrating factor will be this integrating factor will be multiplied by above equation multiply the variable only, leave the constant term after multiplying you will get now we will check it is equal or not so this equation is exact we will integrate it as shown here we will take those terms that do not have x now we will integrate it further so this is our final answer Let us take another question of the same type here, substitute the value of m and n you will see, its value is not equal we will calculate the required value we will simplify it further we will multiply the integrating factor with the above equation after multiplying we will get now we will check it is equal or not it will come as equal then we will directly do its integration and here we will do integration of that term that does not have x term we will integrate it further so in this way we will solve it In reducible to exact we are left with one last method if there is any equation like this if the term is x then it's integrating factor is and if by chance the term is y then it's integrating factor is Students I want to tell you that there is sometimes confusion that how we will know which method to use amongst the two so look carefully, here, m is bigger and n is smaller the one which is smaller will come in the denominator so we don't have problem while solving here m is smaller than n so here we will put formula of 1/m so let us see how to solve this question we will calculate the required value then we will calculate the value of now we will calculate the difference we will solve it as follows so the value will be now we will calculate its integrating factor and integrating factor should be in x only we will simplify it further we will multiply it further simplifying it as you can see these two values are not equal these two are equal now we will do its integration we will solve it s follows so this is our final answer let us move on to next question required values of m and n we will calculate this value as m is smaller here, so we will use this formula observe it carefully this is the value of m we will simplify it further we have done its differentiation keep on simplifying it so its value will be then we will calculate its integrating factor we will multiply the equation with this value as shown after multiplying the final equation will be let us do its integration so students we discussed exact differential equation and we discussed three questions then we learned about reducible to the exact equation and in that we discussed if it is homogeneous how to solve it we learned another method in which x or y is coming common then this will be its required integrating factor we discussed questions related to that method as well I hope my videos are beneftting you as I am getting a positive response do regular comments and do like my video and dont forget to subscribe my channel it is really helpful for engineering and BSC students it is also 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