Linear differential equations are of the form
where the differential operator L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side f is a given function of the same nature as y (called the source term). For a function dependent on time we may write the equation more expressly as
and, even more precisely by bracketing
![L [y(t)] = f(t)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tkWJx3Ii8XKPsm6YJg-smkenBXE-ac5_yZ5fkAt5ILu1vAP5SQhwysVJ_vkqDePhe_fKAB08uIvUlnnhZzmrAG9CqillkkjOtmd_VgkX28EqfUwjz-PtZ_tSzki9ve8XDFsSAIH0QpYCufIPkb=s0-d)
The linear operator L may be considered to be of the form[1]
The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y. It is convenient to rewrite this equation in an operator form
![L_n(y) \equiv \left[\,D^n + A_{1}(t)D^{n-1} + \cdots + A_{n-1}(t) D + A_n(t)\right] y](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v3UVQ1wdsNh1x3enB0_67OEozmcxehID1vuJ9T13SEXnlPg1l-iBZb7X7VsqOtmqKy0EgKA7V0HFMslJyOp7eAmUdrl8POBsC0XSt2m-nfmv6Pov3dHPs09igpP8TvP9U8WCjQayxhzJywssX48w=s0-d)
where D is the differential operator d/dt (i.e. Dy = y' , D2y = y",... ), and the An are given functions.
Such an equation is said to have order n, the index of the highest derivative of y that is involved.
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