857 B
857 B
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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