on 08-Nov-2023 (Wed)

The principle of fault modeling is to reduce the number of effects to be tested by considering how defects manifest themselves.

The order of a differential equation is the highest derivative within the equation.

The Linear Ordinary Differential Equation is the ordinary differential equation which has no non-linear terms like \(y^{2},y^{3},\cdots,y^{\prime 2},y^{\prime 3},\cdots,yy^{\prime},\cdots,y^{\prime}y^{\prime\prime},\cdots\).

The Non-Linear Ordinary Differential Equation is the ordinary differential equation which has at least one non-linear terms like \(y^{2},y^{3},\cdots,y^{\prime 2},y^{\prime 3},\cdots,yy^{\prime},\cdots,y^{\prime}y^{\prime\prime},\cdots\)

Depending on whether the ODE has non-linear terms, the classification of the ODE are Linear ODE and Non-linear ODE.

The general Form for \(2^{nd}\) Order Linear ODE is \(a(x)\frac{d^{2}y}{dx^{2}}+b(x)\frac{dy}{dx}+c(x)y=d(x)\) where \(a(x),b(x),c(x),d(x)\) are different function of \(x\).

The homogeneous ODE is the ODE which has no terms with a pure function of \(x\).

The inhomogeneous ODE is the ODE which has at least one term with a pure function of \(x\).

Depending on whether it has a pure function of non-variable function \(x\), the classification of the ODE is the homogeneous ODE(Having no non-variable function) and inhomogeneous ODE(Having at least one non-variable function).

If an ODE is linear and homogeneous, then the "Principle of Superposition" appears: If \(y_{1}\) and \(y_{2}\) both solve an ODE, then, \(y_{3}=\alpha y_{1}+\beta y_{2}\) is also a solution.

The general form of \(2^{nd}\) order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\) where \(a_{n}(x),\ n=0,1,2\) are the function of \(x\).

欧拉公式是复分析领域的公式，它将三角函数与复指数函数关联起来，因其提出者莱昂哈德·欧拉而得名

欧拉公式提出，对任意实数\(x\)，都存在\(e^{ix} = \cos x + i\sin x\)，其中\(e\)是自然对数的底数，\(i\)是虚数单位，而\(\cos\)和\(\sin\)则是余弦、正弦对应的三角函数，参数\(x\)则以弧度为单位

由于该公式在\(x\)为复数时仍然成立，所以也有人将这一更通用的版本称为欧拉公式：\(e^θ = \cosh θ + \sinh θ\)

The general form of 2nd-order linear homogeneous ODE is \(a_{2}(x)\frac{d^{2}y}{dx^{2}}+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y(x)=0\)
If \(a_{2}(x),a_{1}(x),a_{0}(x)\) are constant functions:
1. Try ansatz: \(y=e^{mx}\).
2. To find \(m\), plug \(y = e^{mx}\) into ODE. Rearrange the equation and get the auxiliary equation: \(\underbrace{(am^{2}+bm+c)}_{\text{Auxiliary Equation}}e^{mx}=0\)
3. Using the solution of the auxiliary equation to solve the ODE.

Depending on the solution of the auxiliary equation, the solution of the ODE is different: \(\displaystyle m=\frac{-b\pm\Delta}{2a}, \Delta\equiv b^{2}-4ac\)
1. \(\Delta>0\Rightarrow\) 2 real roots: \(m_{1},m_{2}\), both \(y_{1}=e^{m_{1}x},y_{2}=e^{m_{2}x}\) works. So the general solution is \(y_{general}=Ae^{m_{1}x}+Be^{m_{2}x}\)
- Alternative form: Because of the Euler's Equation in \(\mathbb{C}\), \(y(x)=e^{px}[\tilde{A}\cosh(qx)+\tilde{B}\sinh(qx)]\), where \(p,q\) are \(\displaystyle m=\underbrace{- \frac{b}{2a}}_p\pm \underbrace{\frac{\sqrt{\Delta}}{2a}}_{q}\).
- Why we need this form? When the initial condition includes \(y(0)\), the alt form would be much easier
2. \(\Delta<0\Rightarrow\) 2 complex roots: \(m_{1},m_{2}\in\mathbb{C}\), so the general solution is \(y(x)=Ce^{m_{1}x}+De^{m_{2}x}\)
- Alternative form: Because of the Euler's Equation, \(y(x)=e^{m_{r}x}[\tilde{C}\cos(m_{i}x)+\tilde{D}\sin(m_{i}x)]\), where \(m_{1,2}=m_{r}\pm im_{i}\).
3. \(\Delta=0\Rightarrow\) one \(m\) only! so the general solution is \(y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}\)