Q. 4

# In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.y = e2x (a + bx)

It is given y = e2x(a + bx) -------(1)

Now, differentiating both side w.r.t. x, we get,

y’ = 2e2x(a + bx) + e2x.b ------(2)

Now, let us multiply equation (1) with 2 and then subtracting it to equation (2), we get,

y’ – 2y = e2x(2a +2bx + b) – e2x(2a + 2bx)

y’ – 2y = be2x ---------(3)

Now, again differentiating both sides w.r.t. x, we get,

y” – 2y’ = 2be2x ------(4)

Dividing equation (4) by equation (3), we get,

y” – 2y’ = 2y’ – 4y

y” – 4y’ – 4y = 0

Therefore, the required differential equation is y” – 4y’ - 4y = 0.

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