# Differential Equation : Find General Solution

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Find a general solution on (-pi/2, pi/2) to y'' + y = tan x given that S secx dx = ln |sec x + tan x|.

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#### Solution Preview

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(i) This equation is in the standard form: y'' + y = tanx

(ii) The associated homogenous equation is y'' + y = 0

The characteristic equation is: Î»2 + 1 = 0

Î»1 = i and Î»2 = -i

yh = c1 cos x + c2 sin x

y1 = cos x, y2 = sin x, g(x) = tanx

y1' = -sin x, y2' = cos x

(iii)

(iv)

(v)

(vi) yp = u1y1 + u2y2 = cos x(sin x - ln (secx + tanx)) - sin x cos x

= -cos x ...

#### Solution Summary

A general solution is found to a differential equation. The solution is detailed and well-presented.

$2.19