3.1 Homogeneous DEs with constant coefficients

Definition

Many second-order ordinary differential equations have the form

$$\frac{d^2y}{dt^2} = f(t ,y ,\frac{dy}{dt})$$

It is linear if the function has the form

$$\frac{d^2 y}{dt^2} = g(t)- p(t)\frac{dy}{dt} - q(t) y$$

In general,

$$y'' + p(t)y' + q(t)y = g(t)$$ $$P(t)y'' + Q(t)y' + R(t)y = 0$$

if $g(t) = 0 \Longrightarrow \text{homogeneous}$

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Initial Value Problem(IVP)

we consider that

$$\begin{cases} ay'' + by' +cy = 0\\ \quad \\ y(0) = y_0 ,\quad y'(0) = y_0'\\ \end{cases}$$

We start by seeking exponential solutions of the form$y = e^{rt}$,where r is a parameter to be determined. Then it follows that $y' = re^{rt}$,$y' = r^2 e^{rt}$Substituting these expressions.

$$(a r^2 + b r + c) e^{rt} = 0$$

Since $e^{rt} \neq 0$

$$a r^2 + b r + c = 0$$

This equation is called the characteristic equation for the differential equation

Assuming that the roots of the characteristic equation are real and different, let them be denoted by $r_1$ and $r_2$, where $r_1 \neq r_2$. Then $y_1(t) = e^{r_1 t} \ and \ y_2(t) = e ^{r_2 t}$

$$then ,\exist \ c_1 ,c_2 \in R \ s.t.\ y = c_1 e^{r_1t} + c_2 e ^{r_2t}$$ $$y' = c_1 r_1 e^{r_1 t} + c_2 r_2 e^{r_2 t}$$ $$y'' = c_1 r_1^2 e^{r_1 t} + c_2 r_2^2 e^{r_2 t}$$

Substituting these expression,

$$ay'' + by' + c = c_1 (ar_1^2 + b r_1 + c)e^{r_1 t} + c_2 (ar_2^2 + b r_2 + c)e^{r_2 t} = 0$$

Consider the initial value $y(t_0) = y_0 ,\ y'(t_0) = y_0'$

$$\begin{cases}c_1 e^{r_1t_0} + c_2 e ^{r_2t_0} = y_0\\ \\ c_1 r_1 e^{r_1 t_0} + c_2 r_2 e^{r_2 t_0} = y_0'\\ \end{cases}$$ $$c_1 = \frac{y_0'-y_0 r_2}{r_1 - r_2}e^{-r_1t_0},\ c_2 = \frac{y_0r_1 - y_0'}{r_2 - r_1} e^{-r_2 t_0}$$