プロセス工学数学A
Takuya Kawanishi
◀ Previous TOC Next ▶

Linear Ordinary Differential Equations (Review)

  • linear differential equations

Definition of Linear ODE and Strategy for Finding the Solutions

Definition (Linear differential equaions.)

  • \(y\) is a function of \(x\)

    \[y = f(x)\]

    A linear differential equation of order \(n\) （\(n\) 階線形微分方程式） can be written as

    \begin{equation} a_n(x) \frac{d^n y}{d x^n} + \dotsb + a_1(x) \frac{d y}{d x} + a_0(x) y = Q(x) \label{eq: def linear ode} \end{equation}

    You have to remember at least up to \(n=2\),

    \[a_2(x) \frac{d^2 y}{dx^2} + a_1(x) \frac{dy}{dx} + a_0(x) y = Q(x)\]

  • Remark

    • The left side of \eqref{eq: def linear ode} is the linear combination （線型結合） of the elements of the following vector.

      \begin{equation} \left[ \frac{d^n y}{d x^n}, \dotsc, \frac{dy}{dx}, y \right] \label{eq: vector derivatives} \end{equation}

      with the coefficeints \(a_n(x), \dotsc, a_0(x)\)

    • Note that the terms in the terms in \eqref{eq: vector derivatives} are linearly independent （線形独立） in the space of continuous functions.

    • The right side term \(Q(x)\) should not be the function of \(y\), should be the function of \(x\) or variables other than \(y\).

    • We cannot have the terms like

      \[y^2, y \frac{dy}{dx}, \ \text{ or } \ \left(\frac{d^2 y}{d x^2}\right)^{1/2}\]

Classification of Linear Ordinary Differential Equations

  • Homogeneous and nonhomogeneous

    • A linear ODE is said to be homogeneous if

      \[Q(x) = 0.\]

    • A linear ODE is called nonhomogeneous if \(Q(x) \ne 0\).

  • Constant-coefficient and variable-coefficients

    • A linear ODE is called (of) constant-coefficent if

      \[a_i(x) = \text{ constant } \quad \text{ for all } i.\]

    • A linear ODE is called variable-coefficient if

      \[a_i(x) \ne \text{ constant } \quad \text{ for some } i.\]

Review

  • linear combination

  • linearly independent

Quiz 1

  • Note that

    \[y^\prime = \frac{dy}{dx}, \quad y^{\prime\prime} = \frac{d^2 y}{dx^2}, \quad y^{(n)} = \frac{d^n y}{d x^n}\]

  • Linear or nonlinear?

    \[y^\prime + y = 2x \tag{a.1}\]
    \[y^{(5)} + x^2 y^{(3)} + x^{1/2} = 0 \tag{a.2}\]
    \[y^{(3)} + y^{(2)} y + 3x y = 0 \tag{a.3}\]
    \[x y^{(3)} + x^2 y^{(2)} + x^3 y^\prime + x^4 y = 0 \tag{a.4}\]
    \[y^{(2)} + y^\prime + y = - x^2 \tag{a.5}\]
    \[y^{(2)} + y^\prime + y = - y x^2 \tag{a.6}\]
    \[y^{(2)} + y^\prime + y = - y^2 x \tag{a.7}\]
    \[y^{(3)} + y^\prime y + 3 y = -2x^3 \tag{a.8}\]

Solution strategy

coefficients

homegeneity

solution tactics

constant

homogeneous

characteristic function

variable

homogeneous

change of variablespower series expansion

constant

non-homogeneous

variation of constantsundetermined coefficientsoperator method

variable

non-homogeneous

combinations of above

  • Being able to choose the appropriate solution is important not only for solving the equation by hand (analytically) but also for using appropriate numerical methods.

解法の選択

係数

同次非同次

解法

定(数)係数

同次

特性方程式

変(数)係数

同次

変数変換べき級数展開

定(数)係数

非同次

定数変化法未定係数法演算子法

変(数)係数

非同次

上記の組合せ

  • 適切な解法を選べるようになることは, 方程式を手で（解析的に）解くのに重要なだけでなく, 数値的に解くときに適切な計算手法を選択するためにも重要である.

Solution for the first order linear ODE

  • The first order linear ODE can be solved regardless of whether it is of constant or variable coefficients or whether it is homogeneous or nonhomogeneous.

  • The solution of the following first order linear ODE

    \[y^\prime + a(x) y = f(x) \tag{a}\]

    is

    \[y = \frac{1}{\mu(x)}\left( \int \mu(x) f(x) \, dx + C \right) \tag{b}\]

    where \(\mu(x)\) is called the integrating factor and defined as

    \[\mu(x) = \exp \left( \int a(x) \, dx \right) \tag{c}\]

Initial Value Problems and Boundary Value Problems

  • Mathematically, no difference.

  • Convenient way of distinguishing between (a) semi-infinite and (b) bounded systems.

  • Initial value problems are for semi-infinite domain, and

  • boundaary value problems are for bounded system.

How to solve initial value and boundary value problems

  • Find the general solution for the differential equations.

  • Apply the intial or boundary condition(s).


◀ Previous TOC Next ▶