Differences

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--- howto:hambasics:sections:test [2021/01/04 21:55] – [Euler Identity and Polar-Cartesian Representations] va7fi
+++ howto:hambasics:sections:test [2021/01/04 22:00] – [Differential Equations] va7fi
@@ Line 192: / Line 192: @@
 \$$z_1 = 1 + i = \sqrt{2}e^\left(i\frac{\pi}{4}\right) \quad \text{and} \quad z_2 = -1 + i = \sqrt{2}e^\left(i\frac{3\pi}{4}\right) \$$
-Imagine having to add, subtract, multiply, divide or divide these together.  Or raise them to a power, or take a root of them.  Which of the two representations do you think would be easiest to use for each operation?
+Imagine having to add, subtract, multiply, or divide these together.  Or raise them to a power, or take a root of them.  Which of the two representations do you think would be easiest to use for each operation?
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@@ Line 351: / Line 351: @@
 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} \$$