### 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.

Order and degree of an ODE

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#.

\[ y''+4y^2-5^{y'} = 0\]

The degree is #undefined#

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

The term #-5^{y'}# shows that the ODE is not a polynomial equation in the derivatives of #y#. Therefore, the degree is not defined.

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