Fundamental Solutions of the Homogeneous Equation

Suppose p and q in Equation 3.7 are continuous on a x b, then for any twice-differentiable function p on a x b, the linear dijferential operator L is defined to mean 

The use of the operator L here reduces the task of integrating the linear second-order differential equation as will be seen below. However, before the integration process can be employed, a few more definitions are needed. 

Theorem 3.2 is a statement of the superposition principle [1,2,4], which is also applicable to higher-order linear differential equations. The two solutions yi and y2 form what is called a fundamental set of solutions for Equation 3.10. 

In general, two solutions yi and y2 of Equation 3.10 are said to form a fundamental set of solutions if every solution of Equation 3.10 can be expressed as a linear combination of yi and y2- In particular 

The conditicm described by Equatirai 3.11 is called the Wronskian and is commonly written in the determinant form as [1] 

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