Consider the ODES x²y" — 2xy' + (x² + 2)y = 0, x²y" — 2xy' + (x² + 2)y - = X sin(x) (2) (3) Verify that y₁ = xcos(x) is a solution to ODE (2). Find a fundamental set of solutions for the ODE (2), show they are indeed linearly independent, and identify the intervals on the real line where the Wronskian of this set of fundamental solutions is non-zero. Find the general solution of the ODE (3).

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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help me with part c please

Consider the ODES
x²y" — 2xy' + (x² + 2)y = 0,
x²y" — 2xy' + (x² + 2)y
-
=
X
sin(x)
(2)
(3)
Verify that y₁ = xcos(x) is a solution to ODE (2).
Find a fundamental set of solutions for the ODE (2), show they are indeed linearly
independent, and identify the intervals on the real line where the Wronskian of this set of
fundamental solutions is non-zero.
Find the general solution of the ODE (3).
Transcribed Image Text:Consider the ODES x²y" — 2xy' + (x² + 2)y = 0, x²y" — 2xy' + (x² + 2)y - = X sin(x) (2) (3) Verify that y₁ = xcos(x) is a solution to ODE (2). Find a fundamental set of solutions for the ODE (2), show they are indeed linearly independent, and identify the intervals on the real line where the Wronskian of this set of fundamental solutions is non-zero. Find the general solution of the ODE (3).
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