 Hi and welcome to the session. I am Deepika here. Let's discuss the question which says for the following differential equation, find the particular solution that is finally given condition x sin square y over x minus y into dx thus x3 y is equal to 0, y is equal to 4 then x is equal to 1. Let's start the solution. Now the given differential equation is sin square y over x minus y into dx plus x dy is equal to 0 or this can be written as x dy is equal to minus of x sin square y over x minus y into dx or dy by dx is equal to y minus x sin square y over x upon x this can be written as dy by dx is equal to y over x minus sin square y over x. Let us give this equation as number one. Clearly this is a homogeneous differential equation. Now we will solve this differential equation by putting y is equal to vx so put y is equal to vx therefore dy by dx is equal to v plus x till v over dx. Now on substituting the value of y and dy by dx in equation 1 we have so from equation 1 v plus x dv over dx is equal to v minus sin square v x dv over dx is equal to v minus sin square v minus v over x dv over dx is equal to minus sin square v on separating the variables we have dv over minus sin square v is equal to dx over x. Now on integrating both sides we have integral minus sin square v dv is equal to integral of dx over x or cot v is equal to log mod x to the base e plus c. Now on replacing v by y over x we have cot y over x is equal to log mod x plus c. Let us give this equation as number 2. Now we are given the condition y is equal to pi by 4 when x is equal to 1 therefore we have pi by 4 is equal to log 1 to the base e plus c. Now cot pi by 4 is equal to 1 and log 1 is equal to 0 plus c or we can write this equation as c is equal to log e to the base e log e to the base e is equal to 1. On substituting the value of c in equation 2 we get cot y over x is equal to log mod x plus log e y over x is equal to log of mod e x the particular solution given differential equation under the condition y is equal to pi by 4 when x is equal to 1 is cot y over x is equal to log of mod e x this is the answer for the above question. This completes our session I hope the solution is clear to you and you have enjoyed the session. Bye and take care.