Classification of first-order ODEs

Section 1.1 Classification of first-order ODEs

Let \(y\) be a suitably differentiable function of \(t\text{.}\) Recall that derivatives of \(y\) with respect to \(t\) can be written as

\begin{equation*} y'=\dfrac{\mathrm{d}y}{\mathrm{d}t}=\dot{y}=Dy\text{,} \end{equation*}

\begin{equation*} y''=\dfrac{\mathrm{d}^2y}{\mathrm{d}t^2}=\ddot{y}=D^2 y\text{,} \end{equation*}

\begin{equation*} \vdots \end{equation*}

\begin{equation*} y^{(n)}=\dfrac{\mathrm{d}^ny}{\mathrm{d}t^n}=D^n y\text{.} \end{equation*}

Definition 1.1.1.

A differential equation is an equation containing the derivatives of one or more unknown functions or dependent variables, with respect to one or more independent variables. A differential equation is called an ordinary differential equation whenever the unknown function has one independent variable. If the unkonwn function has two or more independent variables, then we call it a partial differential equation.

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Definition 1.1.2.

A first-order ordinary differential equation can be written in the form

\begin{equation} y'=f(t,y),\tag{1.1.1} \end{equation}

where \(t\) is the independent variable and \(y\) is the independent variable.

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Most ODEs can’t be solved by hand. However, we can solve a number of ODEs of special and useful types. Some of these properties will carry over to higher-order ODEs.

Definition 1.1.3. Separable ODEs.

A first-order ODE is said to be separable if it can be written in the form

\begin{equation} y'=f(t,y)=g(t)h(y)\tag{1.1.2} \end{equation}

for some functions \(g\) (dependent only on \(t\)), and \(h\) (dependent only on \(y\)).

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Definition 1.1.4.

A first-order ODE is said to be autonomous if it can be written in the form

\begin{equation} y'=f(y)\text{.}\tag{1.1.3} \end{equation}

In other words, there is no explicit dependence of the right-hand side \(f\) on the time variable \(t\text{.}\)

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Definition 1.1.5.

A first-order ODE is said to be linear if it can be written in the form

\begin{equation} a_1(t)y'+a_0(t)y=g(t),\tag{1.1.4} \end{equation}

where \(g\) is called a forcing function (or input or control). If \(g(t)=0\) for all \(x\text{,}\) then (1.1.4) is called homogeneous. Otherwise, (1.1.4) is called nonhomogeneous.

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Remark 1.1.6.

A few things to note about first-order linear ODEs.

A first-order ODE is linear when \(y\) (or whatever the dependent variable is called) and \(y'\) are both of degree one and the coefficients depend on at most a function of \(t\) (or whatever the independent variable is called).

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(1.1.4) can also be written as
\begin{gather*} a_1(t)y'+a_0(t)y \equiv a_1(t)Dy + a_0(t)y = g(t)\\ \underbrace{a_1(t)D+a_0(t)}_{=:L}y=g(t)\\ Ly=g(t), \end{gather*}
where \(L:=a_1(t) \dfrac{\mathrm{d}}{\mathrm{d}t}+a_0(t)\) is called a differential operator.

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Example 1.1.7.

\(\displaystyle (1-t)y'+7ty=\cos(t)\)

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\(\displaystyle (y-2)y'+4y=9\)

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\(\displaystyle \dfrac{\mathrm{d}A}{\mathrm{d}t} = 10-\dfrac{3A}{140-t}\)

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\(\displaystyle \dfrac{\mathrm{d}z}{\mathrm{d\xi}} + 2\xi z^2=\xi e^{\xi}\)

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\(\displaystyle w'-4e^w=6t\)

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