L Verify that y₁ (t) = 1+t² and y₂(t) = e are solutions of (*) (1+t)y" + (-1+t)y - 2y = 0. What are the general solutions of (*)? Find the solution of (*) under the initial conditions y(0) = 1, y'(0) = 2.
L Verify that y₁ (t) = 1+t² and y₂(t) = e are solutions of (*) (1+t)y" + (-1+t)y - 2y = 0. What are the general solutions of (*)? Find the solution of (*) under the initial conditions y(0) = 1, y'(0) = 2.
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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Question
![[17] (a) Verify that y₁ (t) = 1 + t² and y₂(t) = e are solutions of
(*)
(1+t)y" + (−1+t)y' - 2y = 0.
(b) What are the general solutions of (*)?
(c) Find the solution of (*) under the initial conditions y(0) = 1, y′(0) = 2.
(d) Convert (*) to a system of first order differential equations. What are the general
solutions of the obtained system?
(e) Verify that yo(t) = e2 is a solution of
(**)
(1+t)y" + (−1+t)y - 2y = 6te²¹.
(f) What are the general solutions of (**)?
(g) Find the solution of (**) under the initial conditions y(0) = 7, y'(0) = 5.
(h) Convert (**) to a system of first order differential equations. What are the general
solutions of the obtained system?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c89c353-252e-4eca-b673-064be33ffdb0%2Ffe4d9bcd-c881-44cc-a72c-aebc20ec55d7%2F9qlaq1t_processed.png&w=3840&q=75)
Transcribed Image Text:[17] (a) Verify that y₁ (t) = 1 + t² and y₂(t) = e are solutions of
(*)
(1+t)y" + (−1+t)y' - 2y = 0.
(b) What are the general solutions of (*)?
(c) Find the solution of (*) under the initial conditions y(0) = 1, y′(0) = 2.
(d) Convert (*) to a system of first order differential equations. What are the general
solutions of the obtained system?
(e) Verify that yo(t) = e2 is a solution of
(**)
(1+t)y" + (−1+t)y - 2y = 6te²¹.
(f) What are the general solutions of (**)?
(g) Find the solution of (**) under the initial conditions y(0) = 7, y'(0) = 5.
(h) Convert (**) to a system of first order differential equations. What are the general
solutions of the obtained system?
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