 Good morning friends, I am Purva and today we will work out the following question. For the following differential equation find the particular solution satisfying the given condition and we are given dy by dx minus y upon x plus cosec y by x is equal to 0 and the given condition is y is equal to 0 when x is equal to 1. Let us now begin with the solution. Now the given differential equation is dy by dx minus y upon x plus cosec y by x is equal to 0 or we can write this as dy by dx is equal to y upon x minus cosec y upon x we mark this as equation 1. Now clearly we can see that this equation 1 is a homogeneous differential equation therefore to solve it we put y is equal to vx or we have v is equal to y upon x. Now differentiating both sides with respect to x we get differentiating y we get dy by dx is equal to differentiating vx using product rule we get v plus x into dv by dx we mark this as equation 2. Now putting the value of dy by dx from equation 2 in equation 1 we get v plus x into dv by dx is equal to v minus cosec v because we know that y upon x is equal to v. Now this implies x into dv by dx is equal to v minus cosec v minus v cancelling out v we get this implies x into dv by dx is equal to minus cosec v now this further implies dv upon minus cosec v is equal to dx upon x this implies now 1 upon minus cosec v is minus sine v dv is equal to dx upon x. Now integrating both the sides we get integral of minus sine v dv is equal to integral of dx upon x this implies now integrating left hand side we get cosec v is equal to integrating right hand side we get log of mod x plus c where c is the constant of integration now since v is equal to y upon x so we get this implies cos y upon x is equal to log mod x plus c now we are given a condition that when x is equal to 1 y is equal to 0 so putting the value of x and y in this equation we get cos 0 is equal to log 1 plus c which implies now cos 0 is equal to 1 so we get 1 is equal to now log 1 is equal to 0 so we get c and this implies log e is equal to c since we know that log e is equal to 1 or we get c is equal to log e now putting the value of c in this equation 3 we get cos y upon x is equal to log mod x plus log e and this implies cos y upon x is equal to log mod of e x since we know that log m plus log n is equal to log m n therefore the required solution is cos y upon x is equal to log mod e x this is our answer hope you have understood the solution bye and take care