Question

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = e^−2x (A cos x + B sin x)

Answer 1

y = e^−2x (A cos x + B sin x)

∴ e^2x 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.