10.1 Terminology

• First, second, or higher order differential equation
• Initial vs boundary value problem
• Linear vs. nonlinear differential equation
• Homogenous vs. inhomogeneous equation
• Homogenous vs. particular solution
• Ordinary vs. partial differential solution
• Linear coupled differential equation
• Stiff equation

\[a_n(x)y^{(n)} + an_{n-1}(x)y^{n-1} + \cdots + a_1(x)y' + a_0(x)y - f(x) = 0\] or \[a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x)\frac{dy^{n-1}}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)y = f(x)\]

\[\frac{d\vec{x}}{dt} = A \vec{x} + f(t)\] or \[\frac{d\vec{y}}{dx} = A \vec{y} + f(x)\]

\[(y - x)dx + xdy = 0\] \[y'' - 3y' + y = 0\] \[\frac{d^3y}{dx^3} - x \frac{dy}{dx} + 2y = e^x\] \[(1 - y)y' + 3y = e^x\] \[\frac{d^2y}{dx^2} + cos y = 0\] \[\frac{d^4y}{dx^4} + y^3 = 0\] \[V\frac{dC}{dt}= \frac{V_{max} \cdot C}{K_M + C}\]

\[\begin{align*} u' &= 998u + 1998v \\ v' &= -999u - 1999v \\ \end{align*}\]

Initial condition: \[u(0) = 1, \quad v(0) = 1\]