By Shepley L. Ross
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Additional resources for Differential equations
Cn yn (x0 ) = 0 in c1 , . . , cn . As n > 2, result (b) of the last section implies that there is a solution c1 = C1 , . . , cn = Cn with C1 , . . , Cn not all zero. 38 Chapter 3: The Homogeneous Linear Equation and Wronskians Using Proposition 2 above and Theorem 4 of Chapter 2, we deduce, as in the proof of Proposition 3, that C1 y1 + . . , yn are linearly dependent on [a, b]. We conclude our discussion of the solution of (1) by using Wronskians to show that this linear equation actually possesses two linearly independent solutions.
Proposition 2 If y1 , . . , yn are solutions of (1) and c1 , . . , cn are real constants, then c1 y1 + . . + cn yn is also a solution of (1). We now show that the converse of Proposition 1 holds when y1 and y2 are solutions of (1). The result we prove looks at ﬁrst sight (and misleadingly) stronger. Proposition 3 If y1 , y2 are solutions of the linear equation (1) and if W (y1 , y2 )(x0 ) = 0 for some x0 in [a, b], then y1 and y2 are linearly dependent on [a, b] (and hence W (y1 , y2 ) is identically zero on [a, b]).
We justify this terminology by the following proposition, which shows that, once two linearly independent solutions of (2) are found, all that remains to be done in solving (1) is to ﬁnd one particular integral. 42 Chapter 4: The Non-Homogeneous Linear Equation Proposition 1 Suppose that yP is any particular integral of the non-homogeneous linear equation (1) and that y1 , y2 are linearly independent solutions of the corresponding homogeneous equation (2). Then (a) c1 y1 + c2 y2 + yP is a solution of (1) for any choice of the constants c1 , c2 , (b) if y is any solution (that is, any particular integral) of (1), there exist (particular) real constants C1 , C2 such that y = C1 y1 + C2 y2 + yP .