 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 e raised to the power 2x into a plus bx. Let us now begin with the solution. Now we are given y is equal to e raised to the power 2x into a plus bx. Now we multiply both sides of this equation by e raised to the power minus 2x and we get or y into e raised to the power minus 2x is equal to a plus bx. We mark this as equation 1. Now since the above equation consists of two arbitrary constants a and b, so for eliminating them we shall differentiate equation 1 two times. So differentiating 1 with respect to x we get e raised to the power minus 2x into not differentiating y we get y dash plus y into not differentiating e raised to the power minus 2x we get minus 2 into e raised to the power minus 2x is equal to differentiating a we get 0 plus differentiating bx we get b. This implies e raised to the power minus 2x into y dash minus 2 into e raised to the power minus 2x into y is equal to b. This implies now taking out e raised to the power minus 2x common from left hand side we get e raised to the power minus 2x into y dash minus 2y is equal to b. We mark this as equation 2. Now differentiating equation 2 with respect to x using product rule we get e raised to the power minus 2x into differentiating y dash minus 2y with respect to x we get y double dash minus 2y dash plus y dash minus 2y into now differentiating e raised to the power minus 2x we get minus 2 into e raised to the power minus 2x is equal to differentiating b with respect to x we get 0. This implies e raised to the power minus 2x into y double dash minus 2y dash minus 2 into e raised to the power minus 2x into y dash minus 2y is equal to 0. Now taking out e raised to the power minus 2x common we get this implies e raised to the power minus 2x into y double dash minus 2y dash minus 2y dash plus 4y is equal to 0 and this further implies e raised to the power minus 2x into y double dash minus 4y dash plus 4y is equal to 0 and this implies y double dash minus 4y dash plus 4y is equal to 0 as e raised to the power minus 2x is not equal to 0 and this 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 4y dash plus 4y is equal to 0 this is our answer hope you have understood the solution bye and take care