Find a particular solution Satisfying the given condition y = A cos 2x – B sin 2x

Asked by Aaryan | 1 year ago |  64

##### Solution :-

From the question it is given that,

y = A cos 2x – B sin 2x … [equation (i)]

Now, differentiate the equation (i) with respect x,

$$\dfrac{ dy}{dx}$$= – 2A sin (2x) – 2B cos 2x

Taking common terms outside,

$$\dfrac{ dy}{dx}$$= -2 (A sin 2x + B cos 2x) … [equation (ii)]

Then, the above equation is again differentiating with respect to x we get,

$$\dfrac{ d^2y}{dx^2}$$= – 2 [2A cos 2x – 2B sin 2x]

= -4 [A cos 2x – B sin 2x] … [equation (iii)]

The given differential equation is $$\dfrac{ d^2y}{dx^2}$$ + 4y = 0

Substitute the equation (i) and equation (iii) in given differential equation,

$$\dfrac{ d^2y}{dx^2}$$ + 4y = 0

-4 [A cos 2x – B sin 2x] + 4 (A cos 2x – B sin 2x) = 0

-4A cos 2x + 4B sin 2x + 4A cos 2x – 4B sin 2x = 0

0 = 0

Hence it is verified that, y = A cos 2x – B sin 2x is a solution of the differential equation is $$\dfrac{ d^2y}{dx^2}$$+ 4y = 0.

Answered by Aaryan | 1 year ago

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