Order and degree of an ODE

Differential equations: Introduction to Differential equations

Order and degree of an ODE

Recall that $y^{(n)}$ , the $n$ -th derivative of $y$ , is also called the derivative of order $n$ . The next two notions, order and degree, are used as measures of the size of a differential equation.

The general form of an ordinary differential equation for a function $y$ of a single variable $t$ on a certain interval is

$\varphi(t,y,y',y'',\ldots)=0$ where $\varphi$ is a multivariate function.

The order of this ordinary differential equation is the order of the highest derivative of $y$ occurring in $\varphi$ .
If the function $\varphi$ is a polynomial function in each of the derivatives of $y$ , then the degree of this ordinary differential equation is equal to the degree of $\varphi$ as a polynomial in the highest derivative.

If no derivatives appear in the equation, then it is an ordiinary equation and we agree that the order and degree are both equal to $0$ .

Instead of ODE of order $n$ we also speak of an $n$ -th order ODE.

In determining the degree of an ODE of order $n$ we require that $\varphi(t,y,y',\ldots,y^{(n)})$ is a polynomial in $y'$ and higher derivatives, but we do not require that it is a polynomial in $y$ or in $t$ .

If the order of the ODE is equal to $2$ , then $\varphi$ is a function of four variables: the equation is then of the form $\varphi (t,y,y',y'') = 0$ .

The differential equation

$y''+4\cdot y=0$ is a differential equation of order two because the highest derivative in the differential equation is the second derivative $y''$ of $y$ . The degree of this ODE is equal to $1$ because $\varphi(t,y,y',y'')$ is a polynomial in $y'$ and in $y''$ and because the highest derivative $y''$ only occurs to the first power.

Find the order and degree, if defined, of the differential equation

$(y')^2+4y-\cos^3(y) = 0$

The order is $1$
The degree is $2$

The highest derivative of $y$ occurring in the ODE is ${y' }$ . (It can be found in the term $(y')^2$ .) Therefore, the order is $1$ .

The ODE is a polynomial equation in the derivatives of $y$ , so the degree is defined. The term of highest degree in the ODE, viewed as a polynomial equation in $y'$ , is $(y')^2$ . Therefore the degree is $2$ .

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