Differential equations: Introduction

An equation involving the derivatives of a (univariate) function $y=y(x)$ of $x$ is called an ordinary differential equation. That is, an ordinary differential equation (ODE) is an equation of the form

$$\begin{array}{}\text{(Eq:ode)}& F(x,y,y^{\prime },\cdots ,y^{(n)})=0\end{array}$$

where $F(x,z_0,z_1,\cdots ,z_n)$ is a function of $(n+2)$ variables. If the highest order of the derivatives involved in a differential equation is $n$, then it is called an $n$-th order (ordinary) differential equation.

Example. 

• $3y-x{y}^{\prime }+2(y^{\prime }{)}^2=0$ is a (non-linear) first-order differential equation.
• $3y-x{y}^{\prime }+2y^{″}=0$ is a (linear) second-order differential equation.

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If the function $y=y(x)$ on an interval $I$ satisfies (Eq:ode) for any $x\in I$, that is,

$$F(x,y(x),y^{\prime }(x),\cdots ,y^{(n)}(x))=0,$$

then $y=y(x)$ is said to be a solution of the differential equation (Eq:ode) on $I$.

Example. Consider the second-order differential equation

$$y^{″}+y=0.$$

$y=\sin x$ is a solution of this differential equation on $\mathbb{R}$. □