\[A \frac{d^2y}{dt^2}+B \frac{dy}{dt}+Cy=f(t)\]
is the sum of two parts.\[y_c\]
, called the complementary solution, is the solution to the homogeneous equation \[A \frac{d^2y}{dt^2}+B \frac{dy}{dt}+Cy=0\]
.\[y_p\]
, called the particular solution, is any solution to the non homogeneous equation \[A \frac{d^2y}{dt^2}+B \frac{dy}{dt}+Cy=f(t)\]
.The particular solution must be matched to the function
\[f(t)\]
. The table gives some examples.\[f(t)\] |
\[y_p\] |
| A | B |
\[2+5\] t (or polynomial of degree n) |
\[A+Bt\] (or polynomial of degree n) |
\[D e^{\omega t}, \; \omega \neq \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[Ee^{ \omega t}\] |
\[A e^{\omega t}, \; \omega = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[(C_1+C_2t)e^{ \omega t}\] |
\[Dsin \omega t, \; \omega \neq \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[Esin \omega t +F cos \omega t\] |
\[Dcos \omega t, \; \omega \neq \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[Esin \omega t +F cos \omega t\] |
\[Dsin \omega t, \; \omega = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[e^{\omega t}(Esin \omega t +F cos \omega t )\] |
\[Dcos \omega t, \; \omega = \frac{-B \pm \sqrt{B^2-4AC}}{2A}\] |
\[e^{\omega t}(Esin \omega t +F cos \omega t )\] |