MATHEMATICS 110/E-210, SPRING 2014 VECTOR SPACE METHODS FOR DIFFERENTIAL EQUATIONS Proof list for quiz 1 Last modified: January 25, 2014 On the first quiz on February 19, two of these proofs will appear: one chosen at random from 1-4 and one from 5-7. You can receive one point of extra credit for creating a pdf file of one of these proofs and upoading it to the dropbox on the quiz 1 page before 10 PM Monday, Feb. 17 for posting on the Web site. This offer becomes void for a given proof once three files have been posted. 1. Using variation of parameters, show that the general solution to the firstorder initial value problem l(y) = a(x)y 0 (x) + b(x)y(x) = h(x); y(x0 ) = 0 can be written as the definite integral Z x y(x) = f (x) x0 h(t) dt f (t)a(t) where f (x) is an element of the kernel of l, and confirm this formula by differentiation. 2. Define the Wronskian W (x) for n functions f1 (x), · · · fn (x), and prove that if W (x) is nonzero for any x, the functions are linearly independent. 3. Start with the second-order equation l(y) = a(x)y 00 + b(x)y 0 + c(x)y = 0. Show that the Wronskian W (x) of any two independent solutions f1 (x) and f2 (x) of this equation satisfies the first-order equation a(x)W 0 + b(x)W = 0 and that the solution of this equation is Z W = C exp(− b(x) dx). a(x) 4. Prove that the Laplace transform of y 00 (x) is y˜00 (s) = s2 y˜(s) − sy(0) − y 0 (0). 1 P∞ 00 0 i 5. Start with the equation y = p(x)y + q(x)y, where p(x) = i=0 pi x , and P∞ q(x) = i=0 qi xi . P∞ j Show that, in a solution of the form y(x) = j=0 yj x , the coefficients satisfy (n + 2)(n + 1)yn+2 = n+1 X jpn+1−j yj + n X j=1 qn−j yj . j=0 You do not need to show convergence of the series! 6. Prove that the equation Y 00 (x) = N M 0 Y xY + 1− S (1 − Sx )2 has a solution of the form Y (x) = A(1 − x −λ ) S with λ > 0. Outline the strategy (details not required) for using this equation to prove that the power series solution to the equation y 00 = p(x)y 0 + q(x)y converges wherever the power series for p(x) and q(x) both converge. 7. Consider the equation l(y) = a(x)y 00 + b(x)y 0 + c(x)y = h(x). Suppose that we already know a basis for Ker(l): f1 (x) and f2 (x). R∞ Show that yp (x) = −∞ G(x, t)h(t)dt is a solution to l(yp ) = h(x), where the Green’s function is given by G(x, t) = f2 (x)f1 (t) − f1 (x)f2 (t) [H(x − t) − H(x0 − t)]. a(t)W (t) W (x) is the Wronskian; H(x) is the Heaviside step function. Take your choice: you may either derive the solution using variation of parameters or prove it by differentation. 2

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