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Question
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
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Solution
y = e−2x (A cos x + B sin x)
∴ e2x y = A cos x + B sin x ....(1)
Differentiating twice w.r.t. x, we get
`e^(2x) * dy/dx + y * e^(2x) xx 2 = A(- sin x) + B cos x`
∴ `e^(2x)(dy/dx + 2y) = - A sin x + B cos x`
Differentiating again w.r.t. x, we get
`e^(2x)((d^2y)/dx^2 + 2dy/dx) + (dy/dx + 2y) * e^(2x) xx 2 = - A cos x + B (- sin x)`
∴ `e^(2x)((d^2y)/dx^2 + 2dy/dx + 2dy/dx + 4y) = - (A cos x + B sin x)`
∴ `e^(2x)((d^2y)/dx^2 + 4 dy/dx + 4y) = - e^(2x).y` ....[By (1)]
∴ `(d^2y)/dx^2 + 4 dy/dx + 4y = - y`
∴ `(d^2y)/dx^2 + 4 dy/dx + 5y = 0`
This is the required D.E.
Notes
The answer in the textbook is incorrect.
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