 Good morning friends, I am Poojwa and today we will discuss the following question which of the following differential equations has y is equal to x as one of its particular solutions. a d square y by dx square minus x square into dy by dx plus x into y is equal to x. b d square y by dx square plus x into dy by dx plus x into y is equal to x. y is equal to x c d square y by dx square minus x square into dy by dx plus xy is equal to 0. d d square y by dx square plus x into dy by dx plus xy is equal to 0. Let us now begin with the solution. Now we are given that y is equal to x is one of the solutions of one of the four differential equations. So y is equal to x must satisfy one of the four given differential equations. Now consider y is equal to x and let us mark this as equation one. Now differentiating equation one with respect to x we get. Differentiating y with respect to x we get dy by dx is equal to differentiating x with respect to x we get one. We mark this as equation two. Now differentiating equation two with respect to x we get. Differentiating dy by dx with respect to x we get. d square y by dx square is equal to not differentiating one with respect to x we get 0. And we mark this as equation three. Now we shall substitute the value of y dy by dx and d square y by dx square as obtained above in all the equations a, b, c and d. So equation a is d square y by dx square minus x square into dy by dx plus xy is equal to x. Substituting the values from equation one, two and three we get. d square y by dx square is equal to zero so we have zero minus x square into. Now dy by dx is equal to one so we have into one plus x into. Now y is equal to x so we have x and this is equal to x. This implies zero minus now x square into one is x square plus x into x is x square is equal to x. And this further implies canceling out x square here we get zero is equal to x which is not true. Hence equation one is not a solution of a. Now we consider equation b which is d square y by dx square plus x into dy by dx plus xy is equal to x. Now substituting the values from one, two and three we get. Now d square y by dx square is equal to zero so we have zero plus x into dy by dx is equal to one so we have x into one plus x into. And we have y is equal to x so we get x into x and this is equal to x. Now this implies x into one is equal to x plus x into x is equal to x square is equal to x. And this further implies canceling out x from both the sides we get x square is equal to zero which is not true. Hence one is not a solution of equation b. Now we consider equation c which is d square y by dx square minus x square into dy by dx plus xy is equal to zero. Now substituting the values from equation one, two and three we get zero minus x square into one plus x into x is equal to zero. And this implies minus x square into one is x square plus x into x is x square is equal to zero and this further implies zero is equal to zero which is true. Hence one satisfies c. Now let us consider d and d is d square y by dx square plus x into dy by dx plus xy is equal to zero. Now substituting the values from equation one, two and three we get zero plus x into one plus x into x is equal to zero and this implies x into one is x plus x into x is x square is equal to zero which is not true. Hence one does not satisfy d. Hence the correct option is c. This is our answer. So hope you have understood the solution. Bye and take care.