Solving second order differential equations for forced oscillation

Question

Please refer to this image

I have several doubts regarding the stapes involved in solving this Differential equations

1. Why the $-\delta$ term is interduced in $x_{c}(t)=A e^{i(\omega t-\delta)}$? what is the intuition behind this ? is it still mathematicaly currect all though we have altered the Equation ?

2.How we reached $A(\omega)=\frac{1}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\gamma^{2} \omega^{2}}}$ (Please give some detailed Solution)

Please someone help

Answer 1

1. For regular wave equation you have two basic solutions, one with $\sin$, one with $\cos$. Depending on the initial conditions, the solution is a linear combination of these basis solutions: $$x(t)=A_1\cos(\omega t)+A_2\sin(\omega t)$$ Let's choose $A_1=A\cos\delta$ and $A_2=A\sin\delta$. Here $A=\sqrt{A_1^2+A_2^2}$ and $\tan \delta=A_2/A_1$. Then $$x(t)=A\cos(\omega t-\delta)$$ So all they do is to use an alternative form for the linear combination of the sine and cosine functions.
2. Recall that the requirement for $A$ was that is real. In fact you can also choose it positive. So take the last equation in the first image, and take the absolute value.$$|z|=\sqrt{z\bar z}$$ Note that $|e^{-i\delta}|=1$.