What would the form of the particular integral be of the following differential equation:
$$\frac{d^2y}{dx^2} -4 \frac{dy}{dx} +5y=8 \sin x$$
Should the particular integral be of the form; $$y_p=ax\cos x$$
Because the complementary function is of the form; $$y_c=A\sin x +B\cos x$$
Cheers.
Gurjinder.B
I do apologise in advance if this question seems very obvious.
Best Answer
I would opt for the use of differential operators here.
To find $y_c$ by the following:
$m^2-4m+5=0$
Completing the square gives something as $y_c=2±i$
Hence $y_c=e^{2x}[A\cos x+B\sin x]$
Finding$y_p$ by method of differential operators
$y_p=8[\frac{\sin x}{D^2-4D+5}]$
=$8[\frac{\sin x}{4-4D}]$
Using the conjugate of $4-4D$
=$8[\frac{(4+4D)\sin x}{16-16D^2}]$
=$8[\frac{4\sin x+4\cos x}{32}]$
Hence, $y_p=\sin x+\cos x$
Hence can find general solution.