Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.`y=e^(2x)(a+b x)`

Explanation

Given, `y=e^(2x)(a+bx)`……….`(1)`
`implies (dy)/(dx)=e^(2x)(d)/(dx)(a+bx)+(a+bx)(d)/(dx)e^(2x)`
`implies (dy)/(dx)=e^(2x)(b)+2e^(2x)(a+bx)`
`implies y'=2y+be^(2x)`……`(2)` [from equation `(1)`]
`implies y''=2y'+2be^(2x)`……..`(3)`
Multiply equation `(2)` by `2` and subtracting from equation `(3)`,
`y''-2y'=2y'-4y`
`implies y''-4y'+4y=0`
which is the required differential equation.