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Fun with Differential Operators: For convenience, let a, b, c be constants and D = (a) The differential operators ²D and Dx? are not equal. Find an (operator) equation relating them. (b) The equation (x²D – a)[y] Separation of variables should work. = 0 is the same as x?y' – ay = 0. Find the general solution of this equation. (c) Find the second order linear equation equivalent to (x²D – a)(x² D – b) [y] = 0 (sort of “multiply it out" but bo coroful about m2 D -L Do2). 22 1 22/1 22 : O Find the gonorel golution of such on oguation

Fun with Differential Operators: For convenience, let a, b, c be constants and D = (a) The differential operators ²D and Dx? are not equal. Find an (operator) equation relating them. (b) The equation (x²D – a)[y] Separation of variables should work. = 0 is the same as x?y' – ay = 0. Find the general solution of this equation. (c) Find the second order linear equation equivalent to (x²D – a)(x² D – b) [y] = 0 (sort of “multiply it out" but bo coroful about m2 D -L Do2). 22 1 22/1 22 : O Find the gonorel golution of such on oguation

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Fun with Differential Operators: For convenience, let a, b, c be constants and D = (a) The differential operators ²D and Dx? are not equal. Find an (operator) equation relating them. (b) The equation (x²D – a)[y] = 0 is the same as x?y' – ay = 0. Find the general solution of this equation. Separation of variables should work. (c) Find the second order linear equation equivalent to (x²D – a)(x² D – b) [y] = 0 (sort of "multiply it out" but be careful about x?D + Dx²): ??y"+??y'+??y Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate = 0. Find the general solution of such an equation. roots. (d) Find the third order linear equation equivalent to (x?D - a)3 [y] = 0. Also, find the general solution of this equation.
Transcribed Image Text:Fun with Differential Operators: For convenience, let a, b, c be constants and D = (a) The differential operators ²D and Dx? are not equal. Find an (operator) equation relating them. (b) The equation (x²D – a)[y] = 0 is the same as x?y' – ay = 0. Find the general solution of this equation. Separation of variables should work. (c) Find the second order linear equation equivalent to (x²D – a)(x² D – b) [y] = 0 (sort of "multiply it out" but be careful about x?D + Dx²): ??y"+??y'+??y Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate = 0. Find the general solution of such an equation. roots. (d) Find the third order linear equation equivalent to (x?D - a)3 [y] = 0. Also, find the general solution of this equation.
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