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Math 212a (Fall 2013) Yum-Tong Siu1Green’s Kernel for Sturm-Liouville Equationby Method of Variation of ParametersIn this brief set of notes we consider the following Sturm-Liouville equa-tionLg=fon the interval [a, b] ofR, where the Sturm-Liouville operatorLis of the form(Lg) (x) =ddx(p(x)ddxg(x))−q(x)g(x).Here the two functionsp(x) andq(x) are both infinitely differentiable on [a, b]withp(x)>0 on [a, b] andq(x)≥0 on [a, b]. We will show that a solutiongofLg=fis given byg(x) =∫by=aK(x, y)f(y)dy,where the Green kernelK(x, y) is defined byK(x, y) =φ-(x)φ+(y)Wifa≤x≤y≤b,φ-(y)φ+(x)Wifa≤y≤x≤b.Hereφ−(x) andφ+(x) are twoC-linearly independent solutions of the ho-mogeneous differential equationL(φ) = 0 satisfying the initial conditionsφ−(a) = 0,φ′−(a)̸= 0,φ+(b) = 0,φ′+(b)̸= 0,andW=p(x)φ−(x)φ+(x)φ′−(x)φ′+(x)is the “Wronskian” of twoC-linearly independent solutionsφ−(x) andφ+(x)and is independent ofx.For the special case ofp(x)≡1, we will derive the Green kernel by usingthe “method of variation of parameters”. For the general case we will simplyverify that the Green kernel actually gives a solution. We will assign, as ahomework problem, the derivation of the Green kernel for the general casefrom the “method of variation of parameters”.
Math 212a (Fall 2013) Yum-Tong Siu2Definition of Wronskian.For functionsf1,· · ·, fnon (a, b) the WronskianW(f1,· · ·, fn) is defined byW(f1,· · ·, fn) = det(f(j)k)0≤j≤n−1,1≤k≤n,wheref(j)kis the derivative offkof orderj.Vanishing of Wronskian and Linear Dependency.Iff1,· · ·, fnareR-linearly dependent, thenW(f1,· · ·, fn)≡0. On the other hand, ifW(f1,· · ·, fn−1)is nowhere zero andW(f1,· · ·, fn)≡0, thenf1,· · ·, fnareR-linearly de-pendent.Proof.solve forc1,· · ·, cn−1inf(j)n=∑n−1k=1cjf(j)kfor 0≤j≤n−1 anddifferentiate both sides for 0≤j≤n−2 to show thatc′k≡0 for 1≤k≤n−1.Remark.The conditionW(f1,· · ·, fn−1) being nowhere zero cannot bedropped, as shown in the following counter-example.f1(x) = 1 forx≤0andf1(x) = 1 +x3forx≥0.f2(x) = 1 forx≥0 andf2(x) = 1 +x3forx≤0.f3(x) = 3 +x3.