It was stated (on page 7) that "Norder sets of initial values are especially appropriate for N-order differential equations." The following problems illustrate one reason this is true. In particular, they demonstrate that, if y satisfies some N-order initial-value problem, then it automatically satisfies particular higher-order sets of initial values. Because of this, specifying the initial values for ym) with m 2 N is unnecessary and may even lead to problems with no solutions. a. Assume y satisfies the first-order initial-value problem i. Using the differential equation along with the given value for y(1), determine what value y'(1) must be ii. Is it possible to have a solution to i. Find the value of y(0) and of y(0) ii. Is it possible to have a solution to 3ry+2 with y(1)=2. that also satisfies both y(1)=2 and y'(1)-4? (Give a reason.) iii. Differentiate the given differential equation to obtain a second-order differential equation. Using the equation obtained along with the now known values for y(1) and y'(1), find the value of y(1). iv. Can we continue and find y(1), y) (1)....? b. Assume y satisfies the second-order initial-value problem that also satisfies all of the following: =3zy+z¹ +4-8y=0 with y(0) 3 and y(0) = 5. +44-8y=0 y (0) 3, y(0)-5 and "(0)=0 ?
It was stated (on page 7) that "Norder sets of initial values are especially appropriate for N-order differential equations." The following problems illustrate one reason this is true. In particular, they demonstrate that, if y satisfies some N-order initial-value problem, then it automatically satisfies particular higher-order sets of initial values. Because of this, specifying the initial values for ym) with m 2 N is unnecessary and may even lead to problems with no solutions. a. Assume y satisfies the first-order initial-value problem i. Using the differential equation along with the given value for y(1), determine what value y'(1) must be ii. Is it possible to have a solution to i. Find the value of y(0) and of y(0) ii. Is it possible to have a solution to 3ry+2 with y(1)=2. that also satisfies both y(1)=2 and y'(1)-4? (Give a reason.) iii. Differentiate the given differential equation to obtain a second-order differential equation. Using the equation obtained along with the now known values for y(1) and y'(1), find the value of y(1). iv. Can we continue and find y(1), y) (1)....? b. Assume y satisfies the second-order initial-value problem that also satisfies all of the following: =3zy+z¹ +4-8y=0 with y(0) 3 and y(0) = 5. +44-8y=0 y (0) 3, y(0)-5 and "(0)=0 ?
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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partb
Transcribed Image Text:1.8. It was stated (on page 7) that "Nth-order sets of initial values are especially appropriate for Nth-order differential equations." The following problems illustrate one reason this is true. In particular, they demonstrate that, if y satisfies some Nth-order initial-value problem, then it automatically satisfies particular higher-order
sets of initial values. Because of this, specifying the initial values for ym) with m≥N is unnecessary and may even lead to problems with no solutions.
a. Assume y satisfies the first-order initial-value problem
i. Using the differential equation along with the given value for y(1), determine what value y'(1) must be.
ii. Is it possible to have a solution to
i. Find the value of y" (0) and of y"(0)
ii. Is it possible to have a solution to
=3zy+z² with y(1)=2.
that also satisfies both y(1) = 2 and y'(1)=4? (Give a reason.)
iii. Differentiate the given differential equation to obtain a second-order differential equation. Using the equation obtained along with the now known values for y(1) and y' (1), find the value of y" (1).
iv. Can we continue and find y"(1), y(4) (1), ...?
b. Assume y satisfies the second-order initial-value problem
that also satisfies all of the following:
=3zy +2²
+4-8y=0
with y(0) 3 and y(0) = 5.
+4-8y=0
y (0)=3, y(0)=5 and y" (0)=0 ?
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