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from the differential equation by elimin...

from the differential equation by eliminating the arbitrary constants from the following equations :
`(1) y= e^(x) (A cos x + B sin x)`

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` y =e^(x) ( A cos x + B sin x ) `
` e^(x) .y = A cos x + B sin x `
Differnetiating w.r.t x, we get
` e^(-x) (dy)/(dx) + y.e^(-x) (-1) = A (-sinx ) + B cos x`
` therefore e^(-x) ((dy)/(dx) -y) = A sin x + B cos x`
Differentiating again w.r.t.x, we get ,
`e^(-x) ((d^(2)y)/(dx^(2))-(dy)/(dx))+((dy)/(dx)-y).e^(-x)(-1)=-A cosx + B (- sin x)`
` e^(-x) ((d^(2)y)/(dx^(2))-(dy)/(dx) -(dy)/(dx) +y) = - (A cos x + B sin x) `
`therefore e^(-x) ((d^(2)y)/(dx^(2)) -2(dy)/(dx)+y) = e^(-x) .y`
` (d^(2)y)/(dx^(2)) -2(dy)/(dx) + y = -y therefore (d^(2)y)/(dx^(2)) -2 (dy)/(dx) + 2y =0`
This is the required differential equation.
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