Second-Order Linear DEs

Can always be arranged into the form

y^{″} + p (t) y^{'} + q (t) y = g (t)

where $p, q$ and $g$ are continuous on at least some open interval

With operator notation, we can rewrite the DEs as:

\begin{aligned} L & = D^{2} + p D + q \\ L [y] & = g (t) \end{aligned}

Superposition Principle of Solution to Linear Homogeneous ODEs

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wronskian

Linear Dependence

Homogeneous Linear Second-Order DEs with Constant Coefficients

a y^{″} (x) + b y^{'} (x) + c y (x) = 0

a r^{2} + b r + c = 0

This polynomial is called the characteristic equation of the DE

r = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Case 1: $r_{1}$ and $r_{2}$ are real roots, $r_{1} \neq r_{2}$
$y_{1} = c_{1} e^{r_{1} x}, y_{2} = c_{2} e^{r_{2} x}$

y (x) = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}

Case 2: $r_{1}$ and $r_{2}$ are real roots, $r_{1} = r_{2}$
$b^{2} - 4 a c = 0 \to r_{1} = e^{}$ # Homogeneous Linear Second-Order DEs with Constant Coefficients

a y^{″} (x) + b y^{'} (x) + c y (x) = 0

a r^{2} + b r + c = 0

This polynomial is called the characteristic equation of the DE

r = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Case 1: $r_{1}$ and $r_{2}$ are real roots, $r_{1} \neq r_{2}$
$y_{1} = c_{1} e^{r_{1} x}, y_{2} = c_{2} e^{r_{2} x}$

y (x) = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}

Case 2: $r_{1}$ and $r_{2}$ are real roots, $r_{1} = r_{2}$
$b^{2} - 4 a c = 0 \to r_{1} = e^{}$

Case 3: $r_{1}$ and $r_{2}$ are a complex pair
$r = α \pm i β$

$y = c_{1} e^{α + i β x} + c_{2} e^{α - i β x}$

\begin{array}{r} e^{α + i β x} = e^{α x} e^{i β x} = e^{α x} (\cos (β x) + i \sin (β x)) \\ e^{α - i β x} = e^{α x} e^{i (- β x)} = e^{α x} (\cos (- β x) + i \sin (- β x)) \end{array}

y = e^{α x} [(c_{1} + c_{2}) \cos (β x) + (c_{1} - c_{2}) i \sin (β x)]