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Calculus 2/Non-homogeneous ODE.md

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+---
+type: math
+---
+## Finding particular solutions
+
+### Definitions
+
+- RHS $f(x) = P_{deg}(x) \cdot e^{rx}, p\in \mathbb{R}[x]$
+	- P is a polynomial
+	- Of **first kind**
+	- i.e. $e^{-3x}$; $2x^2+x -3$; $xe^x$
+- RHS $f(x) = e^{rx} \cdot [P_{deg}(x)\\cdot \cos qx + Q_{deg_{2}}(x)\cdot \sin qx]$
+	- P, Q are polynomials
+	- Of **second kind**
+	- i.e. $2\cos x-\sin x$;$x^2e^{-x}\cos 2x$
+- A "constant" is a polynomial of degree 0
+
+##### Hyperbolic sin ($\sinh$)
+Just as a fun fact, it doesn't fit neither of the kinds.
+$$
+\sinh x = \frac{e^x - e^{-x}}{2}
+$$
+### Method of undetermined coeffs
+- RHS of 1st kind
+	- There exists a particular solution of the form
+$$ 
+y_{*}(x) = x^s \cdot R_{m}(x)\cdot e^{rx}
+$$
+	- Where $s \rightarrow^{\text{{def}}} \text{multiplicity } r\in \mathbb{R}$ among the roots of characteristic polynomials for the LHS of the equation