Differential Equation Reducible to Linear form | Lecture-II by GP Sir


Hello Students, I am here with another new topic In the first lecture of the differential equation we studied what is variable separable, In the first-order first degree, we studied variable separable, reducible to variable separable Homogeneous and reducible to homogeneous method so students today we are here with a new topic linear differential equation and after that we will study reducible to a linear differential equation so students what is a linear differential equation and in exam how we will come to know it is linear because it is the biggest problem students face in the exam that it is linear in x or in y so here I want to tell you if it is linear in x what will happen and if it is linear in y what will happen so if it is linear in x and in y so how we will come to know, let us see if it is linear in x then its format is if the differential equation is of this type then it will be linear in x and if it is and if the differential equation is of this type then it will be linear in y so here its integrating factor will be then its answer will be the same type we have it here its integrating factor is and the answer will be here IF is integrating factor so students how we will come to know in exams that it is linear in x or y if in any differential equation, x is coming once and y is coming twice and we have a derivative like this, then it is linear in x and here if y is coming once and x is coming twice I have written 4 examples here first of all, you have to identify it is linear in x or y so we will take the first question here here x is coming once and y is coming twice so it is linear in x look at the second question here also x is coming once and y is coming two times so it is linear in x let us take another question here y is coming once and other terms are x so it is linear in y take another example here y is coming once and other terms are of x, so it is also linear in y so, students, we should be able to identify which question is linear in x and which in y if y is coming once then it is linear in y and if x is coming once then it linear in x now you will have a question if we have 2 times x and 2 times y then what to do? then it is the concept of reducible to a linear equation we will discuss it later so let us see how we will solve linear these are the two methods we discussed to solve you will understand it when we will do questions so, students, we are taking the first question here here x is coming once and y has 2 terms so it is linear in x and if it linear in x then we know we want what we will do here is we have to convert it into dx/dy either cross multiply it or multiply by dx/dy in the next, we will do we want x term here solve as follows so students as you can see the value of p is coming as and value of q as firstly, we have to calculate its integrating factor the formula for IF is after solving we will get now our formula is we will substitute the value for IF we will put this value here Here the integration is important so observe carefully its differentiation will be we will put the value here solve as follows then we will do its integration we will do it by ILATE so here we have taken a question and this question was linear in x as x was coming once then we calculated its P and Q calculated its integrating factor and solved it further so students here we are taking the next question so students here you have to check it is linear in x or y here y is coming once so it is linear in y we have perfect format here so we will solve it further we will calculate Integrating factor its integration will be so this is our integrating factor and it is linear in y so our answer will be solve as follows we will do its integration we will simplify it further this is our answer but you are given a condition here so we will apply this condition what does this mean is So students, in this solution wherever there is y put 0 and where it is x put 1 substituting it the value of c is put the value of c here our final answer will be this is our final answer This type of problem is also known as a boundary value problem as here you are given a condition so that you can have a value of constant so, students, we have discussed 2 questions here the first, if we have linear in x how to solve it second, how to solve it if it is linear in y we have two more questions here you can try these and if you have any problem you can mention it in the comment box now we will talk about reducible to the linear differential equation so students, let us move on to our next topic reducible to linear differential equation so how we will come to know we need to reduce it so look here we discussed what is linear in y differential equation this is linear in y if by chance instead of y there is something like this now it is not linear as y is coming twice so we will reduce it to the linear form what we will do is this term we don’t want so we will divide it whatever the term we have here we will take it as ‘t’ and we will differentiate it we will get the value as this whole equation will be reduced in linear form, so how will it happen, let us see we will put the value here solve as follows so this differential equation is now linear in ‘t’ it is reduced in linear form then we will follow our old process here I will take 2 to 3 examples how we solve such type of equations so, students, we have 3 examples here on reducible to linear form here is our first question there are two y and if we dont have this then it is linear we will divide here after division, it will be reduced what we will do is solve as shown we will get it as as you can see, it is been reduced to linear form so our p is and q is now we will calculate its integrating factor we will differentiate it we will solve it further and get as it is linear in t its answer will be solve as shown and value of q we have as we will integrate and solve so this is the way we will solve it now we will see the next question and use the same concept students you have to remember that we have to do dy/dx or dx/dy always we try to free the bigger bracket first so that we can easily solve it we have to remove dx here so what we will do is we will multiply the y here solve as follows so we will write it as now students have confusion here while changing sides so y has a smaller power so we will tke it on LHS then the value will be now we have to reduce it in the linear form we will divide here what we will get is solve as follows students what we have done here we will do the same here becuase it is the same concept we will take the term as ‘t’ so I am writing here directly 1/y is ‘t’ here then the value is this equation linear here in ‘t’ so now how we will solve it here value of p=1 and q is we will calculate its integrating factor so the value of this is and our answer would be we will use the same concept here here ‘t’ was and will substitute it then we will get the answer as solve as follows This is how we will solve such questions we will take one more last question then we will close this topic here again, the students have confusion that should we do it by dy/dx or dx/dy here we can’t simplify it in the denominator so we have to reciprocate it what we will get is we will multiply it we will take xy on LHS here it is dx/dy in last question, we had dy/dx so we divided it by y so here we will divide it by x to the power 3 we will get it is reducible to linear in x so we will assume ‘t’ as and we will differentiate it we will get the value as we will multiply it with -2 we will get the value as it is reducible to ‘t’ value of p is and q is now we will calculate its integrating factor and we will do its integration and get so our answer will be solve as follows we will assume as ‘t’ take it p instead of t as we have already taken it so what we will get is we will use ILATE here then we will get the value as keep the value of t also we will keep it here and p is so here we will get so this will be our answer so, students, we solved 3 questions on reducible to linear ad I hope you are liking my videos I have already uploaded many videos on engineering mathematics and for BSC students and all those students who are studying mathematics on that, I have uploaded the first video on matrices in which I have discussed I have uploaded videos on Fourier series what is full-range Fourier series etc and also half range Fourier series and Harmon classes and then the next topic was on a differential equation, in which the first video was on first-order first degree in that, we discussed three methods first was the variable separable method reducible to variable separable and homogeneous and this is the second video on differential equation today the first topic was the linear differential equation and we did some questions and discussed reducible to linear form so students, if you are liking my videos also, I am telling you fewer questions, because I think more questions will take more time but if you have a particular question on a particular topic, you can drop a comment I will share more videos on that topic Share your reviews with us Don’t forget to like and share my videos

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