Fun with Differential Operators: For convenience, let a, b, c be constants and D = . (a) The differential operators x²D and Dx² 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)(x²D – b)[y] = but be careful about x²D+ Dx²): ??y"+??y'+??y = 0. Find the general solution of such an equation. Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate (sort of "multiply it out" %3D roots.
Fun with Differential Operators: For convenience, let a, b, c be constants and D = . (a) The differential operators x²D and Dx² 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)(x²D – b)[y] = but be careful about x²D+ Dx²): ??y"+??y'+??y = 0. Find the general solution of such an equation. Consider 3 cases: (1) Distinct real roots, (2) A repeated real root, and (3) A pair of complex conjugate (sort of "multiply it out" %3D roots.
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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Please solve a,c and d parts as soon as possible
![Fun with Differential Operators: For convenience, let a, b, c be constants and D =
(a) The differential operators x²D 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)(x²D – 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfe83958-fab2-41e0-b954-baff63ecab2e%2Ff3c04c45-87c9-47c0-ba77-c20a2f5f7b0f%2Fp8yvp2_processed.png&w=3840&q=75)
Transcribed Image Text:Fun with Differential Operators: For convenience, let a, b, c be constants and D =
(a) The differential operators x²D 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)(x²D – 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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