 Hello and welcome to the session. I am Deepika and I am going to help you to solve the following question. The question says, From the differential equation of the family of curves, y is equal to a cos 2x plus b sin 2x where a and b are constant. So let's start the solution. Now we are given the equation of the family of curves y is equal to a cos 2x plus b sin 2x. So we have y is equal to a cos 2x plus b sin 2x. Let us give this as number one. Now on differentiating both sides of equation one with respect to x, Successively we get dy by dx is equal to minus 2a sin 2x plus 2b cos 2x and d2y over dx square is equal to minus 4a cos 2x. Minus 4b sin 2x or this can be written as d2y over dx square is equal to minus 4 into a cos 2x plus b sin 2x. Let us give this as number two. Now from equation one we have y is equal to a cos 2x plus b sin 2x. So by using one in equation two we have d2y over dx square is equal to minus 4y d2y over dx square plus 4y is equal to 0 and this is free from the arbitrary constants a and b. Hence this is the required differential equation. So this is the answer for the above question. This completes our session. I hope the solution is clear to you. Bye and have a nice day.