Differential equations: Introduction

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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

\[F(x, y, y', \cdots, y^{(n)}) = 0\tag{Eq:ode}\]

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. 

  • \(3y - xy' + 2(y')^2 = 0\) is a (non-linear) first-order differential equation.
  • \(3y - xy' + 2y'' = 0\) 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,
\[F(x, y(x), y'(x), \cdots, y^{(n)}(x)) = 0,\]
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 \(\mathbb{R}\). □


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