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