Particular Solution of Differential Equations While solving second order non-homogenous differential equations of the form
$y''+y'=5$, I realized that unlike while solving the ones of the form $y''+y'+y=Ax^n$ where we assume, $y_p=Ax^n+Bx^{n-1}+\cdots$, we assume in this case that $y_p=Ax$ instead of $A$. Can some one give me some insight. Or if this is wrong, how to find particular solution of this form (even though this is very basic).

The reason is that any constant term will disappear when you differentiate it, so even if you put $y_p=Ax+B$, the term $B$ wouldn't appear at all when you wrote $y''_p + y'_p = 5$ in terms of $x$. Since you're looking for a particular solution, rather than a complete family of solutions, it's fine to drop it.

More generally, if you had
$$a_n\dfrac{d^ny}{dx^n} + a_{n-1} \dfrac{d^{n-1}y}{dx^{n-1}} + \cdots + a_{n-k}\dfrac{d^{k}y}{dx^{k}} = c$$
(or even more general equations than this), where $0 \le k<n$ you'd need only substitute
$$y_p = b_n x^n + \cdots + b_k x^k$$
since terms in $x^{k-1}$ or lower powers of $x$ all disappear when differentiated $\ge k$ times.
