The Wronskian Theorems

The Wronskian Theorems §1. Second order equations. Second Order Wronskian Theorem. Suppose that y1 (t) and y2 (t) are solutions of the seond order linear homogeneous equation L y = 0 on an interval, I. Then, the following are equaivalent. 1. For some t0 ∈ I, det y1 (t0 ) y2 (t0 ) y1′ (t0 ) y2′ (t0 ) 6= 0. 2. For any values a, b there are uniquely determined constants c1 , c2 so that y = c1 y1 + c2 y2 is the solution of, L y = 0, y(t0 ) = a, y ′ (t0 ) = b . 3. Every solution of Ly = 0 is of the form y = c1 y1 + c2 y2 for real constants c1 , c2 . That is, c1 y1 + c2 y2 is the general solution. 4. (Theorem 3.3.1) For all t ∈ I, det y1 (t) y1′ (t) y2 (t) y2′ (t) 6= 0. 5. The solutions y1 and y2 are linearly independent. Remarks. i. The determinant appearing in 1 and 4 is called the wronskian. ii. The equivalence of the first two is Theorem 3.2.2. iii. The equivalence of the third with the first two is Theorem 3.2.4. iv. The equivalence of the fourth with the first three is Theorem 3.3.1. v. The equivalence of the fifth with the first four is Theorem 3.3.3. 1 §2. Equations of order n. Suppose that L = dn dn−1 dn−2 d + a (t) + a (t) + · · · + a (t) + a0 (t) , n−1 n−2 1 dtn dtn−1 dtn−2 dt is an nth order linear differential operators with continuous real valued coefficients on an interval I. Suppose that for j = 1, 2, . . . , n, yj (t) satisfies L yj = 0. Definition. The wronskian of these n solutions is defined as the n × n determinant, y1 (t) y1′ (t) ··· W (t) := det ··· (n−1) (t) y1 y2 (t) y2′ (t) (n−1) y2 (t) ··· ··· ··· yn (t) yn′ (t) ··· . ··· (n−1) yn (t) Definition. The solutions are linearly dependent when there are constants a1 , a2 , . . . , an not all zero so that a1 y1 +a2 y2 +· · ·+an yn = 0. Otherwise they are linearly independent. General Wronskian Theorem. The following are equaivalent. 1. For some t0 ∈ I, W (t0 ) 6= 0. 2. For any values b0 , b1 , . . . , bn−1 there are uniquely determined constants cj so that y = c1 y1 + c2 y2 + · · · + cn yn is the solution of the initial value problem, L y = 0, y(t0 ) = b0 , y ′ (t0 ) = b1 , · · · y (n−1) (t0 ) = bn−1 . 3. Every solution of Ly = 0 is of the form c1 y1 + c2 y2 + · · · + cn yn for real constants cj . That is, c1 y1 + c2 y2 + · · · + cn yn is the general solution. 4. For all t ∈ I, W (t) 6= 0. 5. The solutions y1 , y2 , . . . , yn are linearly independent. The proof is almost identical to the proof in the special case n = 2. 2 §3. First order Systems. Suppose that L = a11 (t)  a21 (t)  A(t) =    d + A(t), dt an1 (t) a12 (t) ... an2 (t) ...  a1n (t) a2n (t)   ,  ann (t) is an first order linear system with continuous coefficients on an interval I.
(j) (j) (j) Suppose that for j = 1, 2, . . . , n, X (j) = x1 (t), x2 (t), . . . , xn (t) satisfies L X = 0. Definition. The wronskian of these n solutions is defined as, h i (1) (1) (1) W (t) := det X (t), X (t), . . . , X (t) , where the n × n matrix on the right has j th column equal to X (j) (t). The wronskian is nonzero at t0 if and only if the vectors X (1) (t0 ), . . . , X (n) (t0 ) are linearly independent, if and only if the vectors X (1) (t0 ), . . . , X (n) (t0 ) form a basis of Rn or Cn depending on the case. General Wronskian Theorem. The following are equaivalent. 1. For some t0 ∈ I, W (t0 ) 6= 0. 2. For any values B ∈ Rn there are uniquely determined constants cj so that X = c1 X (1) +c2 X (2) +· · · +cn X (n) is the solution of the initial value problem, L X = 0, X(0) = B . 3. Every solution of LX = 0 is of the form X = c1 X (1) +c2 X (2) +· · ·+cn X (n) for real constants cj . That is, X = c1 X (1) +c2 X (2) +· · ·+cn X (n) is the general solution. 4. For all t ∈ I, W (t) 6= 0. 5. The solutions X (1) , X (2) , · · · , X (n) are linearly independent. The proof is almost identical to the proof in the special case n = 2. 3