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Macro Math Camp Definitions
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- 1differential equation\[ \dot{x} \left( t \right) \equiv \frac{d x \left( t \right)}{dt} = g \left( x \left( t \right), \, t \right) \]
- 2ordinary differential equation (ode)diff. eq. with a single independent variable
- 3partial differential equationmultivariate diff. eq. function
- 4explicit differential equationdifferential equations of the form \[ x^{\left( n \right)} \left( t \right) = g \left( x^{\left(n-1\right)} \left(t\right),\ldots,\, x \left( t \right), \, t \right), \] where the highest-order derivative is separated from the other terms.
- 5implicit differential equationdifferential equations of the form \[ g \left( x^{\left(n\right)} \left(t\right), \, x^{\left(n-1\right)} \left(t\right), \ldots, \, x \left( t \right), \, t \right) = 0 \]
- 6autonomous differential equationdiff. eq. that does not explicitly depend on time as a separate variable: \[ \dot{x} \left( t \right) = g \left( x \left( t \right) \right) \] otherwise, the diff. eq. is called non-autonomous. only autonomous systems have steady states.
- 7linear differential equationdiff. eq. of the form \[ a_n \left(t\right) x^{\left(n\right)} \left(t\right) + a_{n-1} \left( t\right) x^{\left(n-1\right)} \left(t\right) + \ldots + a_1 \left(t\right) \dot{x} \left(t\right) + a \left(t\right) x \left(t\right) + b \left(t\right) = 0. \] otherwise, the diff. eq. is called non-linear.
- 8homogenous differential equationa linear diff. eq. \[ a_n \left(t\right) x^{\left(n\right)} \left(t\right) + a_{n-1} \left( t\right) x^{\left(n-1\right)} \left(t\right) + \ldots + a_1 \left(t\right) \dot{x} \left(t\right) + a \left(t\right) x \left(t\right) + b \left(t\right) = 0. \] with \( b\left(t\right) = 0 \forall t \).
- 9differential equation of order \(n\)a diff. eq. where the highest derivative appearing is of order \(n\). in general we need \(n\) boundary conditions to get a solution to an ODE of order \(n\).
- 10solution to initial value functiona function \(x: \mathcal{T} \rightarrow \mathbb{R} \) that satisfies \[ x^{\left( n \right)} \left( t \right) = g \left( x^{\left(n-1\right)} \left(t\right),\ldots,\, x \left( t \right), \, t \right), \] for all \(t \in \mathcal{T} \text{ with } x \left(0\right) = x_0 \)
- 11transversality conditionboundary condition that specifies what the value of \(x\left(t\right)\) should be at some time \( T < \infty \text{ or at the limit as } T \rightarrow \infty \).
- 12\(m\)-dimensional system of differential equations\[ \left[ \begin{matrix} x_1^{\left(n\right)}\left(t\right) \\
x_2^{\left(n\right)}\left(t\right) \\
\vdots \\
x_m^{\left(n\right)}\left(t\right) \\ \end{matrix} \right] = \left[ \begin{matrix}
g_1 \left(x^{\left(n-1\right)} \left(t\right), \ldots, \, x \left( t \right), \, t \right) \\
g_2 \left(x^{\left(n-1\right)} \left(t\right) ,\ldots,\, x \left( t \right), \, t \right) \\
\vdots \\
g_m \left(x^{\left(n-1\right)} \left(t\right), \ldots,\, x \left( t \right), \, t \right) \\
\end{matrix} \right] \] - 13first-order ODE system\[ \left[ \begin{matrix}
\dot{x}_1 \left(t\right) \\
\dot{x}_2 \left(t\right) \\
\vdots \\
\dot{x}_m \left(t\right) \\
\end{matrix} \right] = \left[ \begin{matrix}
g_1 \left( x \left(t\right), \, t \right) \\
g_2 \left( x \left(t\right), \, t \right) \\
\vdots \\
g_m \left( x \left(t\right), \, t \right) \\
\end{matrix} \right]
\] - 14—
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- Descripción
- Fichas sobre Macro Math Camp Definitions, creado por Eric Andersen el 24/08/2015.
- Creado por
- Eric Andersen
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