 Hi, and welcome to the session. I am Deepika here. Let's discuss a question which says for the following differential equation Find the particular solution Satisfying the given condition 2x y plus y square minus 2x square D y by Tx is equal to 0 y is equal to 2 when x is equal to 1 So let's start the solution Now the given differential equation is 2x y plus y square minus 2x square D y by Tx is equal to 0 or this can be written as minus 2x square D y by Tx is equal to minus of 2x y plus y square or this can be written as D y by Tx is equal to 2x y plus y square over 2x square Again, we can rewrite this differential equation as D y by Tx is equal to y over x plus 1 over 2 y square over x square. Let us give this equation as number one Now equation one is a homogeneous differential equation to solve this equation we will put y is equal to vx on Substituting both sides with respect to x we have D y by Tx is equal to v plus x D v over Dx Now on substituting the value of y and D y by Tx in equation one So from one we have x D v over Dx equal to v plus 1 over 2 v square or D v over Dx is equal to v plus 1 over 2 v square minus v x D v over Dx is equal to 1 over 2 v square or on separating the variables we have 2 D v over v square is equal to Dx over x Now Integrating both sides we have integral of 2 over v square D v is equal to integral of Dx over x minus 2 over v is equal to log of mod x plus c By replacing v by y over x we have minus 2 x over y is equal to log mod x plus c Let us give this equation as number two Now for the particular solution of the given differential equation we have to find the value of c Now we are given the condition Y is equal to 2 when x is equal to 1 so from equation 2 we have minus 2 into 1 Over 2 is equal to log mod 1 plus c minus 1 is equal to 0 plus c This implies c is equal to minus 1 Now on substituting the value of c in equation 2 we get minus 2 x upon y is equal to log mod x minus 1 or y is equal to 2x upon 1 minus log mod x Clearly y is defined when x is not equal to 0 and x is not equal to e because 1 minus log mod x is equal to 0 implies log mod x is equal to 1 or x is equal to e the particular solution of the given differential equation under the condition y is equal to 2 when x is equal to 1 is y is equal to 2x upon 1 minus log mod x and y is defined when x is not equal to 0 and x is not equal to e So this is our answer for the above question I hope the solution is clear to you and you have enjoyed the session. Bye. Take care