## Linear Initial Value Problems

First and second order linear initial value problems are studied. Initial value problems require ﬁnding a function of time that satisﬁes an equation with derivatives and an initial condition. Examples include the motion of projectiles in Subsection 2.3.3, mixing tanks in Subsection 3.7.4 and population models in Subsection 4.4.3. Additional applications include time dependent models of tuned circuits and resonant mass-spring systems. Particular solutions are found by both undetermined coe!cients and variation of parameters. The system formulations are developed for the second order problems, and numerical solutions are given by the MATLAB dierential equation solvers. Chapters six and seven are an introduction to dierential equations, and this topic is covered in more detail in [1] and [7]

A ﬁrst order linear initial value problem for an unknown function { = {(w) with given i(w) is

g{ gw

= {0 = d{+ i(w) and {(0) = {0=

This can be written in dierential operator form where O({) {0 d{ and O({) = i(w)= The operator O is called linear because

O({+ |) = ({+ |)0 d({+ |) = {0 d{+ |0 d|

O(f{) = (f{)0 d(f{) = f({0 d{) = fO({)=

The objective is to ﬁnd {(w) given d, an initial condition {(0) and the function i(w).