Decide whether or not the method of undetermined coefficient

Decide whether or not the method of undetermined coefficients can be applied to find a particular solution of the given equation. y\"()+3y\'()-y()=sec

Solution

No, The method of undetermined coefficients can not be applied in this given equation.

As this method can only with sine and cosine functions.

Reason: because sine and cosine have a finite family of derivatives, Or in other words derivative of sine and cosine functions are cyclic.

And other trigonometric functions does not have a finite family of derivatives.

Eg

y = sin x, then y\' = cos x, then y\'\' = -sin x, then y\'\'\' = -cosx, then y\'\'\'\' = sinx...........and So on

But in the case of

y = secx, then y\' = secx*tanx, then y\'\' = sec^x*tanx + sec^3 x ....

But it will never give the same derivative.

Let me know in the comments if you have any doubts.