alistairmichael.com/notes/math1052/index.md

title: Solving ODEs 1. Consider the following homogeneous linear second order ODE with constant coefficients: $$ay'' + by' + cy = 0$$ The characteristic equation is $a \lambda ^2 + b\lambda + c = 0$. 2. If $a \lambda ^2 + b\lambda + c = 0$ has two real and distinct roots, $\lambda_1$ and $\lambda_2$, then $y(t) = c_1e^{\lambda_1 t} + c_2e^{\lambda_2t}$. If $\lambda _1 , \lambda _2 < 0$, $y \to 0$ as $t\to \infty$. If $\lambda _1$ or $\lambda _2 > 0$, $y \to \pm \infty$ as $t\to \infty$, depending on the values of the coefficients. 3. If $a \lambda ^2 + b\lambda + c = 0$ has one root, $\lambda$, then $y(t) = (c_1 + c_2t)e^{\lambda t}$. Reduction of order. 4. If $a \lambda ^2 + b\lambda + c = 0$ has complex roots, $a \pm i\beta$, then $y(t) = e^{at}( A \cos\beta t + B\sin(\beta t))$. 5. Now consider the following inhomogeneous linear second order ODE with constant coefficients: $$ay'' + by' + cy = r(x)$$ The general solution is $y = y_H + y_p$, where $y_H$ is the solution to the homogeneous ODE $ay'' + by' + cy' = 0$, and $y_p$ is a particular solution to the inhomogeneous ODE. $y_p$ may be found using the method of underdetermined coefficients. Examples of $r(x)$ | Choice for $y_p$
-----------------------|--------------------------------- $2x + 3$ | $ax + b$
$e^{2x}$ | $Ae^{2x}$
$\sin(x)$ | $A\sin(x) + B\cos(x)$
$\cos(x)$ | $A\sin(x) + B\cos(x)$
$2\sin(x) + 3\cos(x)$ | $A\sin(x) + B\cos(x)$
$xe^x$ | $(ax + b)e^x$
$e^x \sin (2x)$ | $e^x (A\sin(2x) + B\cos(2x))$
$e^{2x} + \sin (x)$ | $Ae^{2x} + B\sin(x) + C\cos(x)$
6. Simple Harmonic Motion & Damped Oscillations ($m$ is the mass of the weight attached to the spring, $k$ is the spring constant, and $\beta$ is the dampinc constant) Simple Harmonic Motion (Undamped): $mx'' + kx = 0$ Damped Oscillation: $mx'' + \beta x' + kx = 0$