## Differential equation – Part II Second-Order ODE

Linear 2nd order ODE with constant coefficients Homogeneous: ${y^{\prime\prime}} + A{y^{\prime}} + By = 0$ Solution: $y = {c_1}{y_1} + {c_2}{y_2}$ in which, ${y_1}$ and ${y_2}$ are two independent solutions of the homogeneous ODE. The basic method to solve this ODE is to try $y = {e^{rt}}$. Plug in and we can get ${r^2}{e^{rt}} + Ar{e^{rt}} + B{e^{rt}} = 0 \to {r^2} + Ar + B = 0$ Case 1: two different real roots $y = {c_1}{e^{{r_1}t}} + {c_2}{e^{{r_2}t}}$ Case

## Differential equation – Part I First-Order ODE

Euler equation: \left\{ \begin{align} & {{x}_{n+1}}={{x}_{n}}+h \\ & {{y}_{n+1}}={{y}_{n}}+{{A}_{n}}h \\ \end{align} \right. \left\{ \begin{align} & h=\text{step}\text{size} \\ & {{A}_{n}}=f\left( {{x}_{n}},{{y}_{n}} \right) \\ \end{align} \right. Error of solution error = exact solution – approximate solution Method 1: smaller step Euler first-order: $e\sim {{c}_{1}}h$, proportional to step size Method 2: find a better slop ${{A}_{n}}$ Heun’s method = improved Euler method = modified Euler method = RK2 \left\{ \begin{align} & {{x}_{n+1}}={{x}_{n}}+h \\ & {{y}_{n+1}}={{y}_{n}}+h\left( \frac{{{A}_{n}}+{{B}_{n}}}{2} \right) \\ \end{align} \right. in which, \({{B}_{n}}=f\left(

## Basic concepts of engineering analysis

Analysis of discrete systems The essence of a lumped-parameter mathematical model is that the state of the system can be described directly with adequate precision by the magnitudes of a finite (and usually small) number of state variables. The solution requires the following steps: System idealization: the actual system is idealized as an assemblage of elements Element equilibrium: the equilibrium requirements of each element are established in terms of state variables Element assemblage: the element interconnection requirements are invoked to

## Finite Element Procedures – Matrix, Vectors, Tensors

Special Matrix symmetric matrix identity matrix / unite matrix symmetric banded matrix the following matrix is a symmetric banded matrix of order 5 and the half-bandwidth is 2. ${\rm{A}} = \left[ {\begin{array}{*{20}{c}} 3&2&1&0&0\\ 2&3&4&1&0\\ 1&4&5&6&1\\ 0&1&6&7&4\\ 0&0&1&4&3 \end{array}} \right]$ diagonal matrix: nonzero elements only on the diagonal of the matrix upper half of the matrix inverse matrix, the inverse of a matrix partitioning of matrix the trace and determinant of a matrix: only defined if the matrix is square tr(A)

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