Question
Find the differential equation of the family of curves y = aex + be2x + ce3x, where a, b. c are arbitrary constants.
Solution
The given equation is
y = aex + be2x + ce3x ...(1)
∴ y1 = (aex + 2be2x – 3ce – 3x) ...(2)
Subtracting (1) from (2), we get,
y1 – y = b e2x – 4 c e–3x ...(3)
∴ y2 – y1 = 2 b e2x + 12 c e–3x ...(4)
Multiplying (3) by 2 and subtracting from (3), are get,
y2 – 3 y1 + 2 y = 20 c e–3x ...(5)
∴ y3 – 3 y2 + 2 y1 = – 60 c e –3x ...(6)
Multiplying (5) by 3 and adding to (6), we get ,
(y3 – 3 y2 + 2 y1) + 3 (y2 – 3 y1 + 2 y) = 0 or y3 – 7 y1 + 6 y = 0
which is required differential equation.
