Linear differential equations
In mathematics, linear differential equations are differential equations having solutions which can be added together in particular linear combinations to form further solutions. They equate 0 to a polynomial that is linear in the value and various derivatives of a variable; its linearity means that each term in the polynomial has degree either 0 or 1.
Linear differential equations can be ordinary (ODEs) or partial (PDEs).
The solutions to (homogeneous) linear differential equations form a vector space (unlike non-linear differential equations).
Linear differential equations are of the form
where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function (called the source term) of the same variable. For a function dependent on time we may write the equation more expressly as
and, even more precisely by bracketing
The linear operator L may be considered to be of the form
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
where D is the differential operator d/dt (i.e. Dy = y' = dy/dt , D2y = y" = d2y/dt2,... ), 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.
A typical simple example is the linear differential equation used to model radioactive decay. Let N(t) denote the number of radioactive atoms remaining in some sample of material at time t. Then for some constant k > 0, the rate at which the radioactive atoms decay can be modelled by
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