The differential equation by eliminating $A, B, C$ from $y = A e^{2 x} + B e^{3 x} + C e^{2 x}$ is

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Solution

Given,

$y = A e^{2 x} + B e^{3 x} + C e^{2 x}$ .........(1)

differentiating on both sides, we get,

$y^{'} = 2 A e^{2 x} + 3 B e^{3 x} + 2 C e^{2 x}$ .........(2)

$(2) - 2 \times (1)$

$y^{'} - 2 y = B e^{3 x}$ .....(3)

differentiating on both sides, we get,

$y^{''} - 2 y^{'} = 3 B e^{3 x}$ .....(4)

$(4) - 3 \times (3)$

$y^{''} - 2 y^{'} - 3 y^{'} + 6 y = 0$

$∴ y^{''} - 5 y^{'} + 6 y = 0$ is the required equation.

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