Additional Information on Linear Equations

In this section a very important fundamental theorem will be discussed. This theorem is important because it resolves two of the issues raised at the end of Section 1.1. Specifically, the theorem addresses existence and uniqueness of a 

An initial value problem for a first-order linear equation will always have a unique solution if the conditions of the theorem stated below are satisfied [1,2]. 

If the functions p and g are continuous on an open interval a x /3 containing the point x = Xq, then there exists a unique function y = f (x) that satisfies the differential equation 

Integrating both sides of Equation 2.32 with respect to x and solving for y 

Further, Since p is continuous for a x /3, it follows that p is defined in this interval and is a nonzero differentiable function. Thus, the conversirai of Equation 2.29 into the form of Equation 2.32 is justified. Also, the functitm //.ghas an antiderivative because p and g are ccmtinuous and Equation 2.33 follows from Equation 2.32. The assumption that there is at least one solution of EquatiOTi 229 is verifiable by substituting Equation 2.33 into Equation 2.29. The initial condition. Equation 2.30, determines the integratimi cmistant c uniquely. 

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