howto:hambasics:sections:test
Differences
This shows you the differences between two versions of the page.
Both sides previous revisionPrevious revisionNext revision | Previous revisionNext revisionBoth sides next revision | ||
howto:hambasics:sections:test [2021/01/04 21:58] – [Cartesian vs Polar] va7fi | howto:hambasics:sections:test [2021/02/13 18:58] – va7fi | ||
---|---|---|---|
Line 24: | Line 24: | ||
\begin{align*} | \begin{align*} | ||
- | (1+i)^2 &= 1 + 2i + i^2 \\ | + | (1+i)^2 |
+ | &= (1+i)\cdot(1+i) \\ | ||
+ | &= 1 + 2i + i^2 \\ | ||
&= 1 + 2i - 1 \\ | &= 1 + 2i - 1 \\ | ||
&= 2i | &= 2i | ||
Line 318: | Line 320: | ||
<WRAP center round info 80%> | <WRAP center round info 80%> | ||
- | A differential equation is an equation that relates a function to its derivatives in some ways and the question is: given some information about the system, what's the function (or family of functions) that satisfy the differential equation. | + | * A differential equation is an equation that relates a function to its derivatives in some ways and the question is: given some information about the system, what's the function (or family of functions) that satisfy the differential equation. |
- | In physics we often use a dot above the function to indicate a derivative with respect to time, where as in math, we'll often use an apostrophe. | + | * In physics we often use a dot above the function to indicate a derivative with respect to time, where as in math, we'll often use an apostrophe. |
\$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x'' | \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x'' | ||
</ | </ | ||
Line 351: | Line 353: | ||
What we've go so far says that our test function \$x(t) = e^{rt}\$ will satisfy the differential equation if \$r\$ is given by above equation. | What we've go so far says that our test function \$x(t) = e^{rt}\$ will satisfy the differential equation if \$r\$ is given by above equation. | ||
- | To simplify the notation, let's define \$\alpha\$ and \$beta\$ as: | + | To simplify the notation, let's define \$\alpha\$ and \$\beta\$ as: |
\$$ \alpha = \dfrac{b}{2a} \qquad \text{and} \qquad \beta = \dfrac{\sqrt{|{b^2 - 4ac}|}}{2a} \$$ | \$$ \alpha = \dfrac{b}{2a} \qquad \text{and} \qquad \beta = \dfrac{\sqrt{|{b^2 - 4ac}|}}{2a} \$$ | ||
Line 358: | Line 360: | ||
\$r\$ then: | \$r\$ then: | ||
- | \$$ r = \left\{ \begin{array}{rl} -\alpha \pm \beta & \text{if } b^2 - 4ac > 0,\\ | + | \$$ r = \left\{ \begin{array}{ll} -\alpha \pm \beta & \text{if } b^2 - 4ac > 0,\\ |
| | ||
howto/hambasics/sections/test.txt · Last modified: 2021/02/13 19:14 by va7fi