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howto:hambasics:sections:test [2021/01/04 21:58] – [Cartesian vs Polar] va7fihowto:hambasics:sections:test [2021/01/06 20:47] – [Differential Equations] va7fi
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 <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.  Physicists don't like the apostrophe too much because they sometimes use it to denote a different coordinate system.  So don't let the notation confuse you:+  * 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.  Physicists don't like the apostrophe too much because they sometimes use it to denote a different coordinate system.  So don't let the notation confuse you:
 \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x''(t) = \frac{d^2x}{dt^2} \$$ \$$ \dot{x}(t) = x'(t) = \frac{dx}{dt} \quad \text{and} \quad \ddot{x}(t) = x''(t) = \frac{d^2x}{dt^2} \$$
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 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.  There is still a lot to unpack however.  For example, since \$r\$ contains a square root, it could be real or complex depending on the values of \$a, b,\$ and \$c\$.  And as we saw above, if \$r\$ is real, then \$x(t)\$ will be a real exponential function.  But if \$r\$ is complex, then we can expect \$x(t)\$ to be some sort of sinusoidal function (recall the Euler Identity). 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.  There is still a lot to unpack however.  For example, since \$r\$ contains a square root, it could be real or complex depending on the values of \$a, b,\$ and \$c\$.  And as we saw above, if \$r\$ is real, then \$x(t)\$ will be a real exponential function.  But if \$r\$ is complex, then we can expect \$x(t)\$ to be some sort of sinusoidal function (recall the Euler Identity).
  
-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} \$$
  
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 \$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,\\ 
  -\alpha \pm i \beta & \text{if } b^2 - 4ac < 0, \end{array} \right. \$$  -\alpha \pm i \beta & \text{if } b^2 - 4ac < 0, \end{array} \right. \$$
  
howto/hambasics/sections/test.txt · Last modified: 2021/02/13 19:14 by va7fi