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Form a differential equation for The family of curve for
y=e2x(a+bx)

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Please find below the solution to the asked query:  

We have,    y = e2xa + bx      .....1Differentiating 1, with respect to x, we get    dydx = e2x × b + a+bx . 2 e2x     ⇒dydx = be2x + 2e2xa+bx   ....2Differentiating 2, with respect to x, we get   d2ydx2 = 2be2x + 2be2x + 2e2xa+bx⇒d2ydx2 = 4be2x + 4e2xa + bx     ......3Multiply 1 by 2 and subtracting it from 2, we getdydx - 2y = be2x + 2e2xa+bx - 2e2xa+bx⇒dydx - 2y = be2x        .....4Now, using 4 and 1 in 3, we getd2ydx2 = 4dydx - 2y  + 4y⇒d2ydx2 = 4dydx - 8y + 4y⇒d2ydx2 = 4dydx - 4y⇒d2ydx2 - 4dydx + 4y = 0

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