# on 09-Nov-2023 (Thu)

#### Flashcard 7597038374156

Question
If an ODE is linear and homogeneous, then the "Principle of Superposition" appears: [The property of the solutions]
If $$y_{1}$$ and $$y_{2}$$ both solve an ODE, then, $$y_{3}=\alpha y_{1}+\beta y_{2}$$ is also a solution.

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Principle of Superposition for ODE's solutions
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.

#### Annotation 7597063802124

 The solution method for Constant-coefficients 2nd order linear homogeneous ODE 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.

#### Flashcard 7597070617868

Question

Depending on the solution of the auxiliary equation, the solution of the corresponding 2nd Linear & homogeneous ODE is different: $$\displaystyle m=\frac{-b\pm\sqrt{\Delta}}{2a}, \Delta\equiv b^{2}-4ac$$

[Answer the normal solutions and their alternative form]

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 $$y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}$$

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The relations between the solution of an ODE and the solution of the ODE's auxiliary equation
Depending on the solution of the auxiliary equation, the solution of the ODE is different: $$m=\frac{-b\pm\Delta}{2a}, \Delta\equiv b^{2}-4ac$$ 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, $$y(x)=e^{m_{r}x}[\tilde{C}\cosh(m_{i}x)+\tilde{D}\sinh(m_{i}x)]$$, where $$m_{1,2}=m_{r}\pm im_{i}$$. 3. $$\Delta=0\Rightarrow$$ one $$m$$ only! so the general solution is $$y(x)=Ae^{mx}+Bxe^{mx}=(A+Bx)e^{mx}$$

#### Annotation 7597079268620

 Definition of Cauchy–Euler equation Cauchy–Euler equation is an non-constant coefficients 2nd order linear homogeneous ODE whose form likes $$x^{2}y''+bxy'+cy=0$$

#### Flashcard 7597081103628

Question

Cauchy–Euler equation is [...]

an non-constant coefficients 2nd order linear homogeneous ODE whose form likes $$x^{2}y''+bxy'+cy=0$$

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Definition of Cauchy–Euler equation
Cauchy–Euler equation is an non-constant coefficients 2nd order linear homogeneous ODE whose form likes $$x^{2}y''+bxy'+cy=0$$

#### Flashcard 7597082676492

Question

[...] is an non-constant coefficients 2nd order linear homogeneous ODE whose form likes $$x^{2}y''+bxy'+cy=0$$

Cauchy–Euler equation

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Definition of Cauchy–Euler equation
Cauchy–Euler equation is an non-constant coefficients 2nd order linear homogeneous ODE whose form likes $$x^{2}y''+bxy'+cy=0$$

#### Annotation 7597085822220

 The "Euler Equation" Method for non-constant-coefficients 2nd order linear homogeneous ODE If the ODE is like or can be transformed to $$x^{2}y''+bxy'+cy=0$$ then it can use the "Euler Equation" method: 1. Try ansatz: $$y=x^{p}$$, $$p$$ is an unknown constant. 2. Plug it in ODE to get the auxiliary equation $$p(p-1)+ap+b=0$$. 3. Using the solution of the auxiliary equation $$p_{1},p_{2}$$ to solve the ODE: $$y_{general}(x)=Ax^{p_{1}}+Bx^{p_{2}}$$ , where $$A,B=$$ constant. - If $$p_{1}=p_{2}=p$$, claim: $$y_{general}(x)=(A+B\ln x)x^{p}$$

#### Flashcard 7597087657228

Question
If the ODE is like or can be transformed to $$x^{2}y''+bxy'+cy=0$$ then it can use the "Euler Equation" method: [...]

1. Try ansatz: $$y=x^{p}$$, $$p$$ is an unknown constant.
2. Plug it in ODE to get the auxiliary equation $$p(p-1)+bp+c=0$$.
3. Using the solution of the auxiliary equation $$p_{1},p_{2}$$ to solve the ODE: $$y_{general}(x)=Ax^{p_{1}}+Bx^{p_{2}}$$ , where $$A,B=$$ constant.
- If $$p_{1}=p_{2}=p$$, claim: $$y_{general}(x)=(A+B\ln x)x^{p}$$

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The solution method for non-constant-coefficients 2nd order linear homogeneous ODE by &quot;Euler Equation&quot; Method
If the ODE is like or can be transformed to $$x^{2}y''+bxy'+cy=0$$ then it can use the "Euler Equation" method: 1. Try ansatz: $$y=x^{p}$$, $$p$$ is an unknown constant. 2. Plug it in ODE to get the auxiliary equation $$p(p-1)+ap+b=0$$. 3. Using the solution of the auxiliary equation $$p_{1},p_{2}$$ to solve the ODE: $$y_{general}(x)=Ax^{p_{1}}+Bx^{p_{2}}$$ , where $$A,B=$$ constant. - If $$p_{1}=p_{2}=p$$, claim: $$y_{general}(x)=(A+B\ln x)x^{p}$$

#### Annotation 7597095259404

 The solution method for non-constant-coefficients 2nd order linear homogeneous ODE For the non-constant coefficients linear homogeneous ODE, there are only two solution methods: - If the ODE is like $$x^{2}y''+bxy'+cy=0$$, then it can be fully solved: "Euler Equation" Method - Otherwise, need "Red of Order"

#### Flashcard 7597097618700

Question
For the non-constant coefficients linear homogeneous ODE, there are only two solution methods: [Answer the method name and their applied condition]
- If the ODE is like $$x^{2}y''+bxy'+cy=0$$, then it can be fully solved: "Euler Equation" Method
For the non-constant coefficients linear homogeneous ODE, there are only two solution methods: - If the ODE is like $$x^{2}y''+bxy'+cy=0$$, then it can be fully solved: "Euler Equation" Method - Otherwise, need "Red of Order"