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# Problem5.3.3 - â€ Problem 5.3.3 Multiplying the ODE through...

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Unformatted text preview: â€¡ Problem 5.3.3 Multiplying the ODE through by H HxL we get But, â€ž2 f â€žf H HxL Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã… + H HxL a HxL Ã…Ã…Ã…Ã…Ã…Ã…Ã… + h HxL 8l b HxL + g HxL< f = 0 â€ž x2 â€žx â€ž2 f â€ž â€žf â€žH â€žf H HxL Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã…Ã… = Ã…Ã…Ã…Ã…Ã…Ã…Ã… JH Ã…Ã…Ã…Ã…Ã…Ã…Ã… N - Ã…Ã…Ã…Ã…Ã…Ã…Ã… Ã…Ã…Ã…Ã…Ã…Ã…Ã… â€ž x2 â€žx â€žx â€žx â€žx Substituting into the previous expression and collecting terms gives â€ž â€žf â€žf â€žH Ã…Ã…Ã…Ã…Ã…Ã…Ã… JH Ã…Ã…Ã…Ã…Ã…Ã…Ã… N + Ã…Ã…Ã…Ã…Ã…Ã…Ã… JH a - Ã…Ã…Ã…Ã…Ã…Ã…Ã… N + Hl b + gL H f = 0 â€žx â€žx â€žx â€žx Hence to be in standard Sturm-Liouville form, we must have â€žh Ã…Ã…Ã…Ã…Ã…Ã…Ã… = a H, or H HxL = C1 Exp Aâ€¡ a HtL â€ž tE â€žx (1) (2) (3) x If we let C1 = 1, then p HxL = H, q HxL = g H, and s HxL = b H (4) (5) ...
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