 Good morning friends, I am Purva and today we will discuss the following question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b and we are given y is equal to a into e raised to the power 3x plus b into e raised to the power minus 2x. Let us now begin with the solution. So we are given y is equal to a into e raised to the power 3x plus b into e raised to the power minus 2x. Let us mark this as equation 1. Now multiplying both the sides of equation 1 by e raised to the power 2x we get or e raised to the power 2x into y is equal to a into e raised to the power 3x into e raised to the power 2x plus b. And this implies e raised to the power 2x into y is equal to a into now e raised to the power 3x into e raised to the power 2x can be written as e raised to the power 5x plus b we mark this as equation 2. Now since the above equation consists of 2 arbitrary constants so for eliminating them we shall differentiate equation 2 2 times. So differentiating equation 2 with respect to x we get e raised to the power 2x into now differentiating y with respect to x we get y dash plus 2 into e raised to the power 2x because differentiating e raised to the power 2x with respect to x gives 2 into e raised to the power 2x into y is equal to 5a into e raised to the power 5x because differentiating e raised to the power 5x gives 5 into e raised to the power 5x plus differentiating b with respect to x gives 0. And we can write this as, this implies e raised to the power 2x into y dash plus 2 into e raised to the power 2x into y is equal to 5a into e raised to the power 5x. Now, multiplying both sides of the above equation by e raised to the power minus 5x, we get e raised to the power minus 5x into e raised to the power 2x into y dash plus 2 into e raised to the power minus 5x into e raised to the power 2x into y is equal to 5a. And this implies, now e raised to the power minus 5x into e raised to the power 2x can be written as e raised to the power minus 3x into y dash plus 2 into again e raised to the power minus 5x into e raised to the power 2x can be written as e raised to the power minus 3x into y is equal to 5a. This implies, now taking out e raised to the power minus 3x common from left hand side we get e raised to the power minus 3x into y dash plus 2y is equal to 5a. We mark this as equation 3. Now, again differentiating equation 3 with respect to x we get e raised to the power minus 3x into not differentiating y dash plus 2y with respect to x we get y double dash plus 2y dash plus y dash plus 2y into differentiating e raised to the power minus 3x with respect to x we get minus 3 into e raised to the power minus 3x is equal to now differentiating 5a we get 0. And this implies e raised to the power minus 3x into y double dash plus 2y dash minus 3 into e raised to the power minus 3x into y dash plus 2y is equal to 0. And this implies now taking out e raised to the power minus 3x common we get e raised to the power minus 3x into y double dash plus 2y dash minus 3y dash minus 6y is equal to 0. And this implies y double dash minus y dash minus 6y is equal to 0 as e raised to the power minus 3x is not equal to 0. And this differential equation does not contain constants a and b. So, we write the above equation does not contain a and b. Hence the required differential equation is y double dash minus y dash minus 6y is equal to 0. This is our answer. Hope you have understood the solution. Bye and take care.