The given equation is
y = ae^x + be^2x + ce^3x    ...(1)
∴ y₁ = (ae^x + 2be^2x – 3ce ^{– 3x})   ...(2)
Subtracting (1) from (2), we get,
y₁ – y = b e^2x – 4 c e^–3x    ...(3)
∴    y₂ – y₁ = 2 b e^2x + 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.