Only the function,\(y\left( t \right)\), and its derivatives are used in determining if a differential equation is linear.
If a differential equation cannot be written in the form, \(\eqref\) then it is called a non-linear differential equation.
Note that the order does not depend on whether or not you’ve got ordinary or partial derivatives in the differential equation.
We will be looking almost exclusively at first and second order differential equations in these notes.
There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.
If an object of mass \(m\) is moving with acceleration \(a\) and being acted on with force \(F\) then Newton’s Second Law tells us.
\[y'\left( x \right) = - \frac\hspacey''\left( x \right) = \frac\] Plug these as well as the function into the differential equation.
\[\begin4\left( \right) 12x\left( \right) 3\left( \right) & = 0\ 15 - 18 3 & = 0\ 0 & = 0\end\] So, \(y\left( x \right) = \) does satisfy the differential equation and hence is a solution.
\[\begin F = ma \label \end\] To see that this is in fact a differential equation we need to rewrite it a little.
First, remember that we can rewrite the acceleration, \(a\), in one of two ways.