 Hi, and welcome to the session. Let's work out the following question the question says Solve the following differential equation that is x squared into dy by dx is equal to 2 x y plus y squared So let us see the solution to this question Here we are given x squared into dy by dx is equal to 2 x y plus y squared this implies dy by dx is equal to 2 x y plus y squared divided by x squared Now we see that it is a homogeneous differential equation So we can write it further as we can divide the numerator and the denominator by x squared and we get dy by dx is equal to 2 y by x Is equal to y squared. This is plus y squared by x squared divided by x squared by x squared becomes 1 now what we do is that y v equal to vx then dy by dx will be equal to v plus x dv by dx For putting this in equation 1 we get v plus x into dv by dx is equal to 2 v plus v square This implies x into dv by dx is equal to 2 v plus v squared minus v This is equal to v square Plus v So this implies dx upon x is equal to dv divided by v into 1 plus v This can be written as dx by x is equal to 1 upon v minus 1 upon 1 plus v into dv. Now integrating both the sides we get t-grill dx upon x is equal to integral 1 upon v minus 1 upon 1 plus v dv This implies log mod x plus log of some constant say c is equal to log of mod v divided by 1 plus v This implies v upon 1 plus v is equal to plus minus x into c This implies now we put back the value of v is y by x and we get y divided by x into 1 plus y by x is equal to plus minus x into c therefore y is equal to plus minus c x into x plus y So this is our answer to this question. I hope that you understood the solution and enjoyed the session. Have a good day