 Good morning friends. I am Purva and today we will work out 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 x into a cos x plus b sin x Let us now begin with the solution So the given equation is Y is equal to e raised to the power x into a cos x plus b sin x Now by multiplying both the sides by e raised to the power minus x we can write the above equation as or Y into e raised to the power minus x is equal to a cos x plus b sin x and we mark this as equation 1 Now since the above equation consists of two arbitrary constants so for eliminating them We shall differentiate equation one two times. So differentiating one with respect to x We get e raised to the power minus x into y dash plus y into minus 1 into e raised to the power minus x is equal to minus a sin x plus b cos x Now taking out e raised to the power minus x common from left hand side we get or e raised to the power minus x into y dash minus y is equal to minus a sin x plus b cos x And we mark this as equation 2 Now again differentiating Equation to with respect to x Using product rule we get e raised to the power minus x Into not differentiating y dash minus y with respect to x we get y double dash minus y dash plus y dash minus y into Now differentiating e raised to the power minus x with respect to x we get minus e raised to the power minus x is equal to minus a cos x minus b sin x and this implies e raised to the power minus x into y double dash minus y dash Minus e raised to the power minus x into y dash minus y is equal to minus of a cos x plus b sin x and This implies now taking out e raised to the power minus x common from left hand side we get e raised to the power minus x into y double dash minus y dash minus y dash plus y is Equal to now on right hand side We know that a cos x plus b sin x is equal to y into e raised to the power minus x from equation one So we write minus y into e raised to the power minus x and this is by equation one and This implies e raised to the power minus x into y double dash Minus 2 y dash plus y plus y into e raised to the power minus x is equal to Zero we can also write this as this implies e raised to the power minus x into y double dash minus 2 y dash plus y plus y is equal to zero and This further implies e raised to the power minus x into y double dash Minus 2 y dash plus 2 y is equal to zero and This implies y double dash minus 2 y dash Plus 2 y is equal to zero as e raised to the power minus x is not equal to zero and this above 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 2 y dash plus 2 y is equal to zero This is our answer. Hope you have understood the solution. Bye and take care