7/26/2019 Ode Problem

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ODE problem

March 8, 2015

1 Solution of iv)

From the reasoning from the paragraph before Example 17.1 (page 117) wededuce that we can conclude that y1(x), . . . , yn(x) are linearly dependent on Jfrom the linear dependence at a single point x0, but only if we can check thaty1(x), . . . , yn(x) all solve the same ODE (17.8).

We can check that they solve the ODE (17.8) by a little linear algebra.The hypothesis tells us that the matrix which defines W(y1, . . . , yn)(x) has

rankn1 for any x. Denote r1(x), . . . , rn(x) the rows of this matrix, so exactlyn 1 of them will be linearly independent, for any fixed x.

We show that the first n 1 ones: r1(x), . . . , rn1(x) are linearly indepen-dent, for any x. Indeed, if they were linearly dependent atx0 then they wouldform a matrix of rank n 2 and thus W(y1, . . . , yn1)(x0) = 0, contradictingthe hypothesis.

It then follows thatrn(x) is a linear combination ofr1(x), . . . , rn1(x)) (with

coefficients that can depend on x. This is exactly the ODE (17.8) (with nreplaced by n 1).

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