 Hello and welcome to the session. I am Deepika here. Let's discuss a question which says, show that the given differential equation is homogeneous and solid. y dash is equal to x plus y over x. What is a homogeneous differential equation? Now, a differential equation of the form dy by dx is equal to f of xy If dy by dx is equal to g of y over x where g of y over x is a homogeneous then the differential equation is called the homogeneous differential equation. So this is the key idea behind our question. We will take the help of this key idea to solve the above question. Let's start the solution. Now, the given differential equation is y dash is equal to x plus y over x or dy by dx is equal to x plus y over x dy by dx is equal to 1 plus y over x. Now, we have expressed our differential equation of the form dy by dx is equal to g of y over x. Let us give this an above equation of the form g of y over x. It is a homogeneous function of degree 0. Therefore, the given differential equation is a homogeneous differential equation. We will solve this homogeneous differential equation. For this, we will put y is equal to vx. Now, on differentiating both sides with respect to x we get dy by dx is equal to v plus x into dv by dx. Now, to solve the above question, our first step was to make the right hand side of the given differential equation a function of y over x. Now, next we put y is equal to vx. Now, we will substitute the value of y and dy by dx in the given differential equation. So, on substituting the value of y and dy by dx in equation 1, we get v plus x into dv over dx is equal to 1 plus v because y is equal to vx. So, this is 1 plus vx over x which is 1 plus v or we can say x into dv over dx is equal to 1 plus v minus v or x into dv over dx is equal to 1. Now, on separating the variables we have dv is equal to dx over x. Now, integrating both sides we have integral of dv is equal to integral of dx over x or v is equal to log of mod x plus c. Now, on replacing v by y over x we get y over x is equal to log of mod x plus c. y is equal to x log mod of x plus cx. The required general solution of the given differential equation is y is equal to x log mod x plus cx. So, this is the answer for the above question. I hope the solution is clear to you and you have enjoyed the session. Bye and take care.