solution of Suppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y₁(t) be a y"+p(t)y'+q(t)y = 0 that has no zeroes on (a, b), and let to € (a, b). (a) (b) (1) Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any other linearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q. (Hint: You may express the solution y2(t) as a definite integral between to and t of the other given functions. You may simplify your final formula by (i) leaving out any additive terms that are multiples of y1(t), and (ii) removing any complicated constant factors.) Under which assumptions is this formula valid? Check that the given y₁ together with the y2 derived in part (a) form a fundamental set of solutions for (1).
solution of Suppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y₁(t) be a y"+p(t)y'+q(t)y = 0 that has no zeroes on (a, b), and let to € (a, b). (a) (b) (1) Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any other linearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q. (Hint: You may express the solution y2(t) as a definite integral between to and t of the other given functions. You may simplify your final formula by (i) leaving out any additive terms that are multiples of y1(t), and (ii) removing any complicated constant factors.) Under which assumptions is this formula valid? Check that the given y₁ together with the y2 derived in part (a) form a fundamental set of solutions for (1).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
for part a . please STRICTLY follow the hint provided in part a . express y2(t) in terms of y1, p , AND q. express it as a deinfite
please ALSO answer the quesiton in part a "under which assumption is the resulting formula for y2 valid. "
this is NOT. a graded quesiton
there is no other information needed. all of the information are provided.
Transcribed Image Text:1.
solution of
Suppose the functions p(t) and q(t) are continuous on the interval I = (a,b). Let y1(t) be a
y" + p(t)y' + q(t)y = 0
(1)
that has no zeroes on (a, b), and let to € (a, b).
(a)
(b)
Assuming y₁(t) is a given solution to the ODE (1), derive a formula for any other
linearly independent solution y2(t) for (1) in terms of the solution y₁ and the coefficients p, q.
(Hint: You may express the solution y2(t) as a definite integral between to and t of the other given
functions. You may simplify your final formula by (i) leaving out any additive terms that are
multiples of y1(t), and (ii) removing any complicated constant factors.)
Under which assumptions is this formula valid?
Check that the given y₁ together with the y2 derived in part (a) form a fundamental set
of solutions for (1).
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