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#.
\[ y'''+3\cdot (y'')^2-7y'+x\cdot y = 0\]
The degree is #1#
The highest derivative of #y# occurring in the ODE is #{y''' }#. (It can be found in the term #y''' #.) Therefore, the order is #3#.
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''' #. Therefore the degree is #1#.
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